Every continuous function $f:ℝ\to ℝ$ can be presented as a sum of a certain power series. I've encountered a statement that I am unable to prove or disprove:


  Every continuous function $f:ℝ\to ℝ$ can be presented as a sum of a certain power series.


Is this statement true? If yes, can somebody prove it?
This isn't even true for smooth functions. The classic example: try to expand 
$$
f(x)=\begin{cases}e^{-1/x^2}&amp;x\ne 0\\
0&amp;x=0\end{cases}
$$
in a power series at $x=0$. Convince yourself that $f^{n}(0)$ exists for any $n$ and is zero (these are the coefficients of a hypothetical power series). But the function certainly isn't zero in any positive radius of the origin.
This is not true in all cases. Power series are differentiable almost everywhere but there exist continuous functions which are differentiable nowhere. Now consider the continuous smooth function $f(x)=\exp(-1/x^2)$ with $f(0)=0$. What power series would you like this to be equal to? You should find that the only power series it could be is: $0 + 0x + 0x^2 +\cdots$ but this function is not zero.