Massieu function

In mathematics, the approximate limit is a generalization of the ordinary limit for real-valued functions of several real variables.

A function f on ${\mathbb{ℝ}}^k$ has an approximate limit y at a point x if there exists a set F that has density 1 at the point such that if x_n is a sequence in F that converges towards x then f(x_n) converges towards y.

Properties

The approximate limit of a function, if it exists, is unique. If f has an ordinary limit at x then it also has an approximate limit with the same value.

We denote the approximate limit of f at x₀ by $\underset{x\to x_0}{\lim }\mathrm{ap}f(x).$

Many of the properties of the ordinary limit are also true for the approximate limit.

In particular, if a is a scalar and f and g are functions, the following equations are true if values on the right-hand side are well-defined (that is the approximate limits exist and in the last equation the approximate limit of g is non-zero.)

$\begin{array}{rl}\underset{x\to x_0}{\lim }\mathrm{ap}a\cdot f(x)& =a\cdot \underset{x\to x_0}{\lim }\mathrm{ap}f(x)\\ \underset{x\to x_0}{\lim }\mathrm{ap}(f(x)+g(x))& =\underset{x\to x_0}{\lim }\mathrm{ap}f(x)+\underset{x\to x_0}{\lim }\mathrm{ap}g(x)\\ \underset{x\to x_0}{\lim }\mathrm{ap}(f(x)-g(x))& =\underset{x\to x_0}{\lim }\mathrm{ap}f(x)-\underset{x\to x_0}{\lim }\mathrm{ap}g(x)\\ \underset{x\to x_0}{\lim }\mathrm{ap}(f(x)\cdot g(x))& =\underset{x\to x_0}{\lim }\mathrm{ap}f(x)\cdot \underset{x\to x_0}{\lim }\mathrm{ap}g(x)\\ \underset{x\to x_0}{\lim }\mathrm{ap}(f(x)/g(x))& =\underset{x\to x_0}{\lim }\mathrm{ap}f(x)/\underset{x\to x_0}{\lim }\mathrm{ap}g(x)\end{array}$

Approximate continuity and differentiability

If

$\underset{x\to x_0}{\lim }\mathrm{ap}f(x)=f(x_0)$

f is said to be approximately continuous at x₀. If f is function of only one real variable and the difference quotient

$\frac{f(x_0+h)-f(x_0)}{h}$

has an approximate limit as h approaches zero we say that f has an approximate derivative at x₀. It turns out that approximate differentiability implies approximate continuity, in perfect analogy with ordinary continuity and differentiability.

It also turns out that the usual rules for the derivative of a sum, difference, product and quotient have straightforward generalizations to the approximate derivative. There is no generalization of the chain rule that is true in general however.

External links

• Approximate continuity at Encyclopedia of Mathematics
• Approximate derivative at Encyclopedia of Mathematics
• Approximate differentiability at Encyclopedia of Mathematics

References

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