Example of function that cannot be expressed as a power series

Question

I want to find an example of a $\textbf{continuous everywhere}$ function but can't be expressed as a convergent power series $\sum_{n=0}^{\infty}c_n(x-a)^n$ near a point $a$ (i.e. on $(a−\epsilon,a+\epsilon)$ for all $\epsilon$, a is constant).

Do I need to consider some Fourier series?

Answer 1

A classical example is the function $f$ defined as

$$f(x)=\begin{cases}e^{-1/x^2}&,x\ne 0\\\\0&,x=0\end{cases}$$

All of the derivatives of $f$ vanish at $0$, and so its Taylor series is $0$. But the remainder term in the Taylor expansion of $f$ is $f$ itself.

Hence, the Taylor series of $f$, namely $0$, does not converge to $f$.

Answer 2

Your constraints allow very simple examples such as $|x|$ or $\sqrt[3]{x}$, both of which cannot be expressed as a power series in $x$ in an interval about $0$.