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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?

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    Answers available here: http://math.stackexchange.com/questions/207217/power-series-of-a-function – 2017-02-11
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    Marking that as a duplicate leaves out the very simple answer of @Hurkyl below. – 2017-02-11

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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 $f$.

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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 $0$.

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    Nice, that answers the question in the simplest way. – 2017-02-11