What is an Example of an Infinitely Differentiable Function that is Nowhere Analytic?

In the realm of mathematical functions, one of the most fascinating and counterintuitive concepts is the existence of infinitely differentiable functions (smooth functions) that are nowhere analytic. An example of such a function is Fe[ This function, while smooth and having derivatives of all orders, does not possess a Taylor series expansion that converges to it within any open interval.

Properties of the Fe[ Function

Infinitely Differentiable: The function Fe[ is ∞-times differentiable (infinitely differentiable) everywhere on the real line. For x ≠ 0, it is a composition of smooth functions, and at x 0, we can show that all derivatives exist and are equal to 0. This is demonstrated by the definition:

Fe(x){e-1x2-1/x2ifx≠00ifx0

Nowhere Analytic

A function is said to be analytic at a point if it can be represented by a power series that converges to the function in some neighborhood of that point. For the function Fe[, the Taylor series centered at x 0 is:

Fe0 Fe02x2 Fe03x3 …f_0 frac{f_0}{2!}x^2 frac{f_0}{3!}x^3 ldots

This series converges to 0 for all x, meaning that the Taylor series does not converge to Fe[ for x ≠ 0. Therefore, Fe[ is not analytic at any point, and hence it is nowhere analytic.

Conclusion

The function Fe[ defined as Fe[ e^{-1/x^2} for x ≠ 0 and Fe[ 0 for x 0 serves as a perfect example of an infinitely differentiable function that is nowhere analytic.

Other Examples and Generalizations

Another approach to creating an infinitely differentiable function that is nowhere analytic is by considering a function that is analytic everywhere except at a dense subset of points. One such function is:

g(x)∑n1∞Fe(x-rn)2^ng(x) sum_{n1}^infty frac{Fe(x-r_n)}{2^n}

Where g(x) is the sum of scaled down versions of the function Fe[ This function g will be infinitely differentiable but does not have a Taylor series expansion that converges to it in any interval. The rational numbers Q are dense within the real numbers, and scaling ensures that the function does not converge to the original function at any rational point.

Conclusion and Final Thoughts

This discussion highlights the existence of infinitely differentiable functions that are nowhere analytic, providing a deeper understanding of the nuances in mathematical analysis. These functions challenge our intuition about the relationship between smoothness and analyticity and are essential for advanced studies in mathematics and related fields.