|
|
| Line 1: |
Line 1: |
| {{otheruses|Oscillation (differential equation)}}
| | The writer's title is Christy Brookins. To climb is something she would never give up. North Carolina is the place he enjoys most but now he is contemplating other choices. Distributing production is exactly where her primary earnings comes from.<br><br>Feel free to surf to my web-site cheap psychic readings ([http://skillroar.com/index/?p=39307 click here! skillroar.com]) |
| | |
| [[File:LimSup.svg|right|thumb|300px|Oscillation of a sequence (shown in blue) is the difference between the limit superior and limit inferior of the sequence.]]
| |
| | |
| In [[mathematics]], '''oscillation''' quantifies the amount that a sequence or function tends to move between extremes. There are several related notions: oscillation of a [[sequence]] of [[real number]]s, oscillation of a real valued [[function (mathematics)|function]] at a point, and oscillation of a function on an [[interval (mathematics)|interval]] (or [[open set]]).
| |
| | |
| ==Definitions==
| |
| ===Oscillation of a sequence===
| |
| If (<math>a_n</math>) is a sequence of real numbers, then the oscillation of is defined as the difference (possibly ∞) between the [[limit superior and limit inferior]] of <math>a_n</math>:
| |
| :<math>\omega(a_n) = \lim\sup a_n - \lim\inf a_n.</math>
| |
| It is undefined if both are +∞ or both are −∞, that is, if the sequence tends to +∞ or to −∞. The oscillation is zero if and only if the sequence converges.
| |
| | |
| ===Oscillation of a function on an open set===
| |
| Let <math>f</math> be a real-valued function of a real variable. The oscillation of <math>f</math> on an interval <math>I</math> in its domain is the difference between the [[supremum]] and [[infimum]] of <math>f</math>:
| |
| :<math>\omega_f(I) = \sup_{x\in I} f(x) - \inf_{x\in I} f(x).</math>
| |
| More generally, if <math>f:X\to\mathbb{R}</math> is a function on a [[topological space]] <math>X</math> (such as a [[metric space]]), then the oscillation of <math>f</math> on an [[open set]] <math>U</math> is
| |
| :<math>\omega_f(U) = \sup_{x\in U} f(x) - \inf_{x\in U}f(x).</math>
| |
| | |
| ===Oscillation of a function at a point===
| |
| The oscillation of a function <math>f</math> of a real variable at a point <math>x_0</math> is defined as the limit as <math>\epsilon\to 0</math> of the oscillation of <math>f</math> on an <math>\epsilon</math>-neighborhood of <math>x_0</math>:
| |
| :<math>\omega_f(x_0) = \lim_{\epsilon\to 0} \omega_f(x_0-\epsilon,x_0+\epsilon).</math>
| |
| This is the same as the difference between the limit superior and limit inferior of the function at <math>x_0</math>, ''provided'' the point <math>x_0</math> is not excluded from the limits.
| |
| | |
| More generally, if <math>f:X\to\mathbb{R}</math> is a real-valued function on a [[metric space]], then the oscillation is
| |
| :<math>\omega_f(x_0) = \lim_{\epsilon\to 0} \omega_f(B_\epsilon(x_0)).</math>
| |
| | |
| ==Examples==
| |
| [[File:Rapid Oscillation.svg|thumb|300px|right|As the argument of ''ƒ'' approaches point ''P'', ''ƒ'' oscillates from ''ƒ''(a) to ''ƒ''(b) infinitely many times, and does not converge.]]
| |
| *1/''x'' has oscillation ∞ at ''x'' = 0, and oscillation 0 at other finite ''x'' and at −∞ and +∞.
| |
| *sin (1/''x'') (the [[topologist's sine curve]]) has oscillation 2 at ''x'' = 0, and 0 elsewhere.
| |
| *sin ''x'' has oscillation 0 at every finite ''x'', and 2 at −∞ and +∞.
| |
| *The sequence 1, −1, 1, −1, 1, −1, ... has oscillation 2.
| |
| | |
| In the last example the sequence is [[Frequency|periodic]], and any sequence that is periodic without being constant will have non-zero oscillation. However, non-zero oscillation does not usually indicate periodicity.
| |
| | |
| Geometrically, the graph of an oscillating function on the real numbers follows some path in the ''xy''-plane, without settling into ever-smaller regions. In [[well-behaved]] cases the path might look like a loop coming back on itself, that is, periodic behaviour; in the worst cases quite irregular movement covering a whole region.
| |
| | |
| == Continuity ==
| |
| Oscillation can be used to define [[continuous function|continuity of a function]], and is easily equivalent to the usual ''ε''-''δ'' definition (in the case of functions defined everywhere on the real line): a function ƒ is continuous at a point ''x''<sub>0</sub> if and only if the oscillation is zero;<ref>''[http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF Introduction to Real Analysis],'' updated April 2010, William F. Trench, Theorem 3.5.2, p. 172</ref> in symbols, <math>\omega_f(x_0) = 0.</math> A benefit of this definition is that it ''quantifies'' discontinuity: the oscillation gives how ''much'' the function is discontinuous at a point.
| |
| | |
| For example, in the [[classification of discontinuities]]:
| |
| * in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
| |
| * in a jump discontinuity, the size of the jump is the oscillation (assuming that the value ''at'' the point lies between these limits from the two sides);
| |
| * in an essential discontinuity, oscillation measures the failure of a limit to exist.
| |
| | |
| This definition is useful in [[descriptive set theory]] to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ''ε'' (hence a [[G-delta set|G<sub>δ</sub> set]]) – and gives a very quick proof of one direction of the [[Lebesgue integrability condition]].<ref>''[http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF Introduction to Real Analysis],'' updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177</ref>
| |
| | |
| The oscillation is equivalence to the ''ε''-''δ'' definition by a simple re-arrangement, and by using a limit ([[lim sup]], [[lim inf]]) to define oscillation: if (at a given point) for a given ''ε''<sub>0</sub> there is no ''δ'' that satisfies the ''ε''-''δ'' definition, then the oscillation is at least ''ε''<sub>0</sub>, and conversely if for every ''ε'' there is a desired ''δ,'' the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.
| |
| | |
| ==Generalizations==
| |
| More generally, if ''f'' : ''X'' → ''Y'' is a function from a [[topological space]] ''X'' into a [[metric space]] ''Y'', then the '''oscillation of ''f''''' is defined at each ''x'' ∈ ''X'' by
| |
| | |
| :<math>\omega(x) = \inf\left\{\mathrm{diam}(f(U))\mid U\mathrm{\ is\ a\ neighborhood\ of\ }x\right\}</math>
| |
| | |
| == See also ==
| |
| * [[Grandi's series]]
| |
| * [[Bounded mean oscillation]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| {{refbegin}}
| |
| *{{cite book|author=Hewitt and Stromberg|title=Real and abstract analysis|page=78|publisher=Springer-Verlag|year=1965}}
| |
| *{{cite book|author=Oxtoby, J|title=Measure and category|publisher=Springer-Verlag|edition=4th ed.|year=1996|pages=31–35|isbn=978-0-387-90508-2}}
| |
| *{{cite book
| |
| | last = Pugh
| |
| | first = C. C.
| |
| | title = Real mathematical analysis
| |
| | publisher = New York: Springer
| |
| | date = 2002
| |
| | pages = 164–165
| |
| | isbn = 0-387-95297-7
| |
| }}
| |
| {{refend}}
| |
| | |
| [[Category:Real analysis]]
| |
| [[Category:Limits (mathematics)]]
| |
| [[Category:Sequences and series]]
| |
| [[Category:Functions and mappings]]
| |
The writer's title is Christy Brookins. To climb is something she would never give up. North Carolina is the place he enjoys most but now he is contemplating other choices. Distributing production is exactly where her primary earnings comes from.
Feel free to surf to my web-site cheap psychic readings (click here! skillroar.com)