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C^infty Function


C-InfinityFunction

A C^infty function is a function that is differentiable for all degrees of differentiation. For instance, f(x)=e^(2x) (left figure above) is C^infty because its nth derivative f^((n))(x)=2^ne^(2x) exists and is continuous. All polynomials are C^infty. The reason for the notation is that C-k have k continuous derivatives.

C^infty functions are also called "smooth" because neither they nor their derivatives have "corners," which would make their graph look somewhat rough. For example, f(x)=|x^3| is not smooth (right figure above).

There are special C^infty functions which are very useful in analysis and geometry. For example, there are smooth functions called bump functions, which are smooth approximations to a characteristic function. Typically, these functions require some calculus to show that they are indeed C^infty.

A smooth, non-analytic function

Any analytic function is smooth. But a smooth function is not necessarily analytic. For instance, an analytic function cannot be a bump function. Consider the following function, whose Taylor series at 0 is identically zero, yet the function is not zero:

 f(x)={0   for x<=0; e^(-1/x)   for x>0.
(1)

The function f goes to zero very quickly. One property of smooth functions is that they can look very different at different scales.

The set of smooth functions cannot be made into a Banach space, which makes some problems hard, but instead has the weaker structure of a Fréchet space.


See also

C-k Function, C-infty Topology, Calculus, Differential Topology, Fréchet Space, Partition of Unity, Sard's Theorem

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "C^infty Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/C-InfinityFunction.html

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