Non-analytic smooth function

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

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.

It remains to show that the right-hand side derivative of f at x = 0 is zero.

For the right-hand side derivative of f (n) at x = 0 we obtain with the above limit As seen earlier, the function f is smooth, and all its derivatives at the origin are 0.

The function has a strictly positive denominator everywhere on the real line, hence g is also smooth.

To have the smooth transition in the real interval [a, b] with a < b, consider the function For real numbers a < b < c < d, the smooth function equals 1 on the closed interval [b, c] and vanishes outside the open interval (a, d), hence it can serve as a bump function.

A more pathological example is an infinitely differentiable function which is not analytic at any point.

It can be constructed by means of a Fourier series as follows.

, this function is easily seen to be of class C∞, by a standard inductive application of the Weierstrass M-test to demonstrate uniform convergence of each series of derivatives.

is not analytic at any dyadic rational multiple of π, that is, at any

of real or complex numbers, the following construction shows the existence of a smooth function F on the real line which has these numbers as derivatives at the origin.

[1] In particular, every sequence of numbers can appear as the coefficients of the Taylor series of a smooth function.

With the smooth transition function g as above, define This function h is also smooth; it equals 1 on the closed interval [−1,1] and vanishes outside the open interval (−2,2).

Using h, define for every natural number n (including zero) the smooth function which agrees with the monomial xn on [−1,1] and vanishes outside the interval (−2,2).

Therefore, the constants involving the supremum norm of ψn and its first n derivatives, are well-defined real numbers.

Define the scaled functions By repeated application of the chain rule, and, using the previous result for the k-th derivative of ψn at zero, It remains to show that the function is well defined and can be differentiated term-by-term infinitely many times.

[2] To this end, observe that for every k where the remaining infinite series converges by the ratio test.

For every radius r > 0, with Euclidean norm ||x|| defines a smooth function on n-dimensional Euclidean space with support in the ball of radius r, but

Indeed, all holomorphic functions are analytic, so that the failure of the function f defined in this article to be analytic in spite of its being infinitely differentiable is an indication of one of the most dramatic differences between real-variable and complex-variable analysis.

Note that although the function f has derivatives of all orders over the real line, the analytic continuation of f from the positive half-line x > 0 to the complex plane, that is, the function has an essential singularity at the origin, and hence is not even continuous, much less analytic.

By the great Picard theorem, it attains every complex value (with the exception of zero) infinitely many times in every neighbourhood of the origin.