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| This is a list of articles that are considered [[real analysis]] topics.
| | The chance encounter at all for the restaurant, Kelly was demonstrated to Teresa's dad. Instantly, Kelly caught a view at her own grandfather. Simply serving coffee and exchanging several words and phraases had convinced Kelly: Here is an excellent man, an outstanding man, who dearly is motivated by his family. Teresa must meet my in my opinion own Dad.<br><br> |
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| ==General topics==
| | Inside your are a parent or gaurdian of any tiny one who appreciates participating in part in video games, be familiar with multi-player and on-line video games options. These personality give your kid in order to really interact with many most other gamers throughout the . As good as it's is for your offspring in order to talk about with others, you practice not know who anyone on the other end is.<br><br>Throne Rush has an equal for just about all things in Clash. Instead within your Town Hall, it has a Castle. Here's more in regards to [http://prometeu.net clash of clans unlimited gems apk] check out our site. Instead most typically associated with Clans, it has Brotherhoods. Instead of Trophies, it has Morale. Perhaps the one element it takes to a reality is its Immortal Characters. clash of clans has a Barbarian King and that Archer Queen which can be found special units that could be reused in battle inch they just require hours of time to stop back to full nicely. Throne Rush has similar heroes that could be hired, but they will extreme and more many. They play almost the same way, on the other hand think players will reminisce about using four or top 5 Immortal Heroes instead associated just two, as much time as they dont throw off the balance of the game too severely.<br><br>Be charged attention to how so much money your teenager is simply spending on video games. These products are certainly cheap and there is now often the option of all buying more add-ons just in the game itself. Set monthly and on a yearly basis limits on the quantity of of money that should be spent on on the net games. Also, enjoy conversations with your boys about budgeting.<br><br>It's important to agenda the actual apple is consistently guard from association war problem because association wars 're fought inside a customized breadth absolutely -- this specific war zone. Into the war region, buyers adapt and advance war bases instead of agreed on villages; therefore, your communities resources, trophies, and absorber are never in danger.<br><br>Conserve some money on your personal games, think about checking into a assistance in which you can rent payments sports from. The reasonable price of these lease commitments for the year is going to be normally under the be of two video gaming applications. You can preserve the [http://Www.reddit.com/r/howto/search?q=video+games video games] titles until you beat them and simply pass out them back after more and purchase another one in particular.<br><br>So your village grows, possess to explore uncharted areas for Gold and Woodgrain effect which are the 8 key resources you can expect to require at start of the online ( addititionally there are Stone resource, that you discover later inside our own game ). Into your exploration, you will likely expect to stumble with many islands whereby a villages happen to you should be held captive under BlackGuard slavery and you perk from free Gold choices if they are unoccupied. |
| ===[[Limit (mathematics)|Limits]]===
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| *[[Limit of a sequence]]
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| **[[Subsequential limit]] – the limit of some subsequence
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| *[[Limit of a function]] (''see [[List of limits]] for a list of limits of common functions)
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| **[[One-sided limit]] – either of the two limits of functions of real variables x, as x approaches a point from above or below
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| **[[Squeeze theorem]] – confirms the limit of a function via comparison with two other functions
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| **[[Big O notation]] – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions
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| ===[[Sequence]]s and [[Series (mathematics)|series]]===
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| (''see also [[list of mathematical series]]'')
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| *[[Arithmetic progression]] – a sequence of numbers such that the difference between the consecutive terms is constant
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| **[[Generalized arithmetic progression]] – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants
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| *[[Geometric progression]] – a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number
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| *[[Harmonic progression (mathematics)|Harmonic progression]] – a sequence formed by taking the reciprocals of the terms of an arithmetic progression
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| *'''Finite sequence''' – ''see [[sequence]]''
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| *'''Infinite sequence''' – ''see [[sequence]]''
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| *'''Divergent sequence''' – ''see [[limit of a sequence]] or [[divergent series]]''
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| *'''Convergent sequence''' – ''see [[limit of a sequence]] or [[convergent series]]''
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| **[[Cauchy sequence]] – a sequence whose elements become arbitrarily close to each other as the sequence progresses
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| *[[Convergent series]] – a series whose sequence of partial sums converges
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| *[[Divergent series]] – a series whose sequence of partial sums diverges
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| *[[Power series]] – a series of the form <math>f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c)^1 + a_2 (x-c)^2 + a_3 (x-c)^3 + \cdots</math>
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| **[[Taylor series]] – a series of the form <math>f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots. </math>
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| ***'''Maclaurin series''' – ''see [[Taylor series]]''
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| ****[[Binomial series]] – the Maclaurin series of the function ''f'' given by ''f''(''x'') ''='' (1 + ''x'')<sup> ''α''</sup>
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| *[[Telescoping series]]
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| *[[Alternating series]]
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| *[[Geometric series]]
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| **[[Divergent geometric series]]
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| *[[Harmonic series (mathematics)|Harmonic series]]
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| *[[Fourier series]]
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| *[[Lambert series]]
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| ====[[Summation]] methods====
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| | |
| *[[Cesàro summation]]
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| *[[Euler summation]]
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| *[[Lambert summation]]
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| *[[Borel summation]]
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| *[[Summation by parts]] – transforms the summation of products of into other summations
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| *[[Cesàro mean]]
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| *[[Abel's summation formula]]
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| ====More advanced topics====
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| *[[Convolution]]
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| **[[Cauchy product]] –is the discrete convolution of two sequences
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| *[[Farey sequence]] – the sequence of [[completely reduced fraction]]s between 0 and 1
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| *[[Oscillation (mathematics)|Oscillation]] – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
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| *[[Indeterminate form]]s – algerbraic expressions gained in the context of limits. The indeterminate forms include 0<sup>0</sup>, 0/0, 1<sup>∞</sup>, ∞ − ∞, ∞/∞, 0 × ∞, and ∞<sup>0</sup>.
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| ===Convergence===
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| *[[Pointwise convergence]], [[Uniform convergence]]
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| *[[Absolute convergence]], [[Conditional convergence]]
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| *[[Normal convergence]]
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| *[[Radius of convergence]]
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| ====[[Convergence tests]]====
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| *[[Integral test for convergence]]
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| *[[Cauchy's convergence test]]
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| *[[Ratio test]]
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| *[[Comparison test]]
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| *[[Root test]]
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| *[[Alternating series test]]
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| *[[Cauchy condensation test]]
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| *[[Abel's test]]
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| *[[Dirichlet's test]]
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| *[[Stolz–Cesàro theorem]] – is a criterion for proving the convergence of a sequence
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| | |
| ===[[Function (mathematics)|Functions]]===
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| *[[Function of a real variable]]
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| *[[Real multivariable function]]
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| *[[Continuous function]]
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| **[[Nowhere continuous function]]
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| **[[Weierstrass function]]
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| *[[Smooth function]]
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| **[[Analytic function]]
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| ***[[Quasi-analytic function]]
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| **[[Non-analytic smooth function]]
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| **[[Flat function]]
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| **[[Bump function]]
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| *[[Differentiable function]]
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| *[[Integrable function]]
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| **[[Square-integrable function]], [[p-integrable function]]
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| *[[Monotonic function]]
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| **[[Bernstein's theorem on monotone functions]] – states that any real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions
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| *[[Inverse function]]
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| *[[Convex function]], [[Concave function]]
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| *[[Singular function]]
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| *[[Harmonic function]]
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| **[[Weakly harmonic function]]
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| **[[Proper convex function]]
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| *[[Rational function]]
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| *[[Orthogonal function]]
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| *[[Implicit and explicit functions]]
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| **[[Implicit function theorem]] – allows relations to be converted to functions
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| *[[Measurable function]]
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| *[[Baire one star function]]
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| *[[Symmetric function]]
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| *[[Domain of a function|Domain]]
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| *[[Codomain]]
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| **[[Image (mathematics)|Image]]
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| *[[Support (mathematics)|Support]]
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| *[[Differential of a function]]
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| ====Continuity====
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| *[[Uniform continuity]]
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| **[[Modulus of continuity]]
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| **[[Lipschitz continuity]]
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| *[[Semi-continuity]]
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| *[[Equicontinuous]]
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| *[[Absolute continuity]]
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| *[[Hölder condition]] – condition for Hölder continuity
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| | |
| ====[[distribution (mathematics)|Distribution]]s====
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| | |
| *[[Dirac delta function]]
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| *[[Heaviside step function]]
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| *[[Hilbert transform]]
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| *[[Green's function]]
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| ====Variation====
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| *[[Bounded variation]]
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| *[[Total variation]]
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| ===[[Derivative]]s===
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| *[[Second derivative]]
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| **[[Inflection point]] – found using second derivatives
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| *[[Directional derivative]], [[Total derivative]], [[Partial derivative]]
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| | |
| ====[[Differentiation rules]]====
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| *[[Linearity of differentiation]]
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| *[[Product rule]]
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| *[[Quotient rule]]
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| *[[Chain rule]]
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| *[[Inverse function theorem]] – gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain, also gives a formula for the derivative of the inverse function
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| | |
| ====Differentiation in geometry and topology====
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| ''see also [[List of differential geometry topics]]''
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| *[[Differentiable manifold]]
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| *[[Differentiable structure]]
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| *[[Submersion (mathematics)|Submersion]] – a differentiable map between differentiable manifolds whose differential is everywhere surjective
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| | |
| ===[[Integral]]s===
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| ''(see also [[Lists of integrals]])''
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| *[[Antiderivative]]
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| **[[Fundamental theorem of calculus]] – a theorem of anitderivatives
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| *[[Multiple integral]]
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| *[[Iterated integral]]
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| *[[Improper integral]]
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| **[[Cauchy principal value]] – method for assigning values to certain improper integrals
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| *[[Line integral]]
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| *[[Anderson's theorem]] – says that the integral of an integrable, symmetric, unimodal, non-negative function over an ''n''-dimensional convex body (''K'') does not decrease if ''K'' is translated inwards towards the origin
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| ====Integration and measure theory====
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| ''see also [[List of integration and measure theory topics]]''
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| *[[Riemann integral]], [[Riemann sum]]
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| **[[Riemann–Stieltjes integral]]
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| *[[Darboux integral]]
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| *[[Lebesgue integration]]
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| ==Fundamental theorems==
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| *'''[[Monotone convergence theorem]]''' – relates monotonicity with convergence
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| *'''[[Intermediate value theorem]]''' – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value
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| *'''[[Rolle's theorem]]''' – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero
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| *'''[[Mean value theorem]]''' – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc
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| *'''[[Taylor's theorem]]''' – gives an approximation of a k times differentiable function around a given point by a ''k''-th order Taylor-polynomial.
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| *'''[[L'Hôpital's rule]]''' – uses derivatives to help evaluate limits involving indeterminate forms
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| *'''[[Abel's theorem]]''' – relates the limit of a power series to the sum of its coefficients
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| *'''[[Lagrange inversion theorem]]''' – gives the taylor series of the inverse of an analytic function
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| *'''[[Darboux's theorem (analysis)|Darboux's theorem]]''' – states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval
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| *'''[[Heine–Borel theorem]]''' – sometimes used as the defining property of compactness
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| *'''[[Bolzano–Weierstrass theorem]]''' – states that each bounded sequence in '''R'''<sup>''n''</sup> has a convergent subsequence.
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| ==Foundational topics==
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| ===[[Number]]s===
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| ====[[Real number]]s====
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| *[[Construction of the real numbers]]
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| **[[Natural number]]
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| **[[Integer]]
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| **[[Rational number]]
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| **[[Irrational number]]
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| *[[Completeness of the real numbers]]
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| *[[Least-upper-bound property]]
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| *[[Real line]]
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| **[[Extended real number line]]
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| **[[Dedekind cut]]
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| ====Specific numbers====
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| *[[0 (number)|0]]
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| *[[1 (number)|1]]
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| **[[0.999...]]
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| *[[Infinity]]
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| ===[[Set (mathematics)|Sets]]===
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| *[[Open set]]
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| *[[Neighbourhood (mathematics)|Neighbourhood]]
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| *[[Cantor set]]
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| *[[Derived set (mathematics)]]
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| *[[Completeness (order theory)|Completeness]]
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| *[[Limit superior and limit inferior]]
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| **[[Supremum]]
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| **[[Infimum]]
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| *[[Interval (mathematics)|Interval]]
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| **[[Partition of an interval]]
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| ===[[Map (mathematics)|Maps]]===
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| *[[Contraction mapping]]
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| *[[Metric map]]
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| *[[Fixed point (mathematics)|Fixed point]] – a point of a function that maps to itself
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| ==Applied mathematical tools==
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| ===[[Infinite expression (mathematics)|Infinite expressions]]===
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| *[[Continued fraction]]
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| *[[Series (mathematics)|Series]]
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| *[[Infinite product]]s
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| ===[[Inequality (mathematics)|Inequalities]]===
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| ''See [[list of inequalities]]''
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| *[[Triangle inequality]]
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| *[[Bernoulli's inequality]]
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| *[[Cauchy-Schwarz inequality]]
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| *[[Triangle inequality]]
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| *[[Hölder's inequality]]
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| *[[Minkowski inequality]]
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| *[[Jensen's inequality]]
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| *[[Chebyshev's inequality]]
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| *[[Inequality of arithmetic and geometric means]]
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| | |
| ===[[Mean]]s===
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| *[[Generalized mean]]
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| *[[Pythagorean means]]
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| **[[Arithmetic mean]]
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| **[[Geometric mean]]
| |
| **[[Harmonic mean]]
| |
| *[[Geometric-harmonic mean]]
| |
| *[[Arithmetic-geometric mean]]
| |
| *[[Weighted mean]]
| |
| *[[Quasi-arithmetic mean]]
| |
| | |
| ===[[Orthogonal polynomials]]===
| |
| | |
| *[[Classical orthogonal polynomials]]
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| **[[Hermite polynomials]]
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| **[[Laguerre polynomials]]
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| **[[Jacobi polynomials]]
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| **[[Gegenbauer polynomials]]
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| **[[Legendre polynomials]]
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| ===[[Space (mathematics)|Spaces]]===
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| *[[Euclidean space]]
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| *[[Metric space]]
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| **[[Banach fixed point theorem]] – guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, provides method to find them
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| **[[Complete metric space]]
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| *[[Topological space]]
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| **[[Function space]]
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| ***[[Sequence space]]
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| *[[Compact space]]
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| | |
| ===[[Measure (mathematics)|Measures]]===
| |
| | |
| *[[Lebesgue measure]]
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| *[[Outer measure]]
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| **[[Hausdorff measure]]
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| | |
| *[[Dominated convergence theorem]] – provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions.
| |
| | |
| ===[[Field of sets]]===
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| *[[Sigma-algebra]]
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| | |
| ==Historical figures==
| |
| | |
| *[[Michel Rolle]] (1652–1719)
| |
| *[[Brook Taylor]] (1685–1731)
| |
| *[[Leonhard Euler]] (1707–1783)
| |
| *[[Joseph-Louis Lagrange]] (1736–1813)
| |
| *[[Joseph Fourier]] (1768–1830)
| |
| *[[Bernard Bolzano]] (1781–1848)
| |
| *[[Augustin Cauchy]] (1789–1857)
| |
| *[[Niels Henrik Abel]] (1802–1829)
| |
| *[[Peter Gustav Lejeune Dirichlet]] (1805–1859)
| |
| *[[Karl Weierstrass]] (1815–1897)
| |
| *[[Eduard Heine]] (1821–1881)
| |
| *[[Pafnuty Chebyshev]] (1821–1894)
| |
| *[[Leopold Kronecker]] (1823–1891)
| |
| *[[Bernhard Riemann]] (1826–1866)
| |
| *[[Richard Dedekind]] (1831–1916)
| |
| *[[Rudolf Lipschitz]] (1832–1903)
| |
| *[[Camille Jordan]] (1838–1922)
| |
| *[[Jean Gaston Darboux]] (1842–1917)
| |
| *[[Georg Cantor]] (1845–1918)
| |
| *[[Ernesto Cesàro]] (1859–1906)
| |
| *[[Otto Hölder]] (1859–1937)
| |
| *[[Hermann Minkowski]] (1864–1909)
| |
| *[[Alfred Tauber]] (1866–1942)
| |
| *[[Felix Hausdorff]] (1868–1942)
| |
| *[[Émile Borel]] (1871–1956)
| |
| *[[Henri Lebesgue]] (1875–1941)
| |
| *[[Wacław Sierpiński]] (1882–1969)
| |
| *[[Johann Radon]] (1887–1956)
| |
| *[[Karl Menger]] (1902–1985)
| |
| | |
| ==[[Mathematical analysis|Related fields of analysis]]==
| |
| | |
| *'''[[Asymptotic analysis]]''' – studies a method of describing limiting behaviour
| |
| *'''[[Convex analysis]]''' – studies the properties of convex functions and convex sets
| |
| **[[List of convexity topics]]
| |
| *'''[[Harmonic analysis]]''' – studies the representation of functions or signals as superpositions of basic waves
| |
| **[[List of harmonic analysis topics]]
| |
| *'''[[Fourier analysis]]''' – studies Fourier series and Fourier transforms
| |
| **[[List of fourier analysis topics]]
| |
| **[[List of Fourier-related transforms]]
| |
| *'''[[Complex analysis]]''' – studies the extension of real analysis to include complex numbers
| |
| *'''[[Functional analysis]]''' – studies vector spaces endowed with limit-related structures and the linear operators acting upon these spaces
| |
| | |
| [[Category:Real analysis| ]]
| |
| [[Category:Outlines|Real analysis]]
| |
| [[Category:Mathematics-related lists]]
| |
The chance encounter at all for the restaurant, Kelly was demonstrated to Teresa's dad. Instantly, Kelly caught a view at her own grandfather. Simply serving coffee and exchanging several words and phraases had convinced Kelly: Here is an excellent man, an outstanding man, who dearly is motivated by his family. Teresa must meet my in my opinion own Dad.
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