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Concept
Non-analytic smooth function
Summary
In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below.
One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions.
The existence of smooth but non-analytic functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable manifold is fine, in contrast with the analytic case.
The functions below are generally used to build up partitions of unity on differentiable manifolds.
Consider the function
defined for every real number x.
The function f has continuous derivatives of all orders at every point x of the real line. The formula for these derivatives is
where pn(x) is a polynomial of degree n − 1 given recursively by p1(x) = 1 and
for any positive integer n. From this formula, it is not completely clear that the derivatives are continuous at 0; this follows from the one-sided limit
for any nonnegative integer m.
By the power series representation of the exponential function, we have for every natural number (including zero)
because all the positive terms for are added. Therefore, dividing this inequality by and taking the limit from above,
We now prove the formula for the nth derivative of f by mathematical induction. Using the chain rule, the reciprocal rule, and the fact that the derivative of the exponential function is again the exponential function, we see that the formula is correct for the first derivative of f for all x > 0 and that p1(x) is a polynomial of degree 0. Of course, the derivative of f is zero for x < 0.
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