This is likely a very simple and well-known issue (bear in mind my thoughts are rather loose here):
I'm considering the subset of continuous functions spanned by the space of exponential functions, i.e. functions of the form
$$ \sum_{- \infty}^{\infty} a_j e^{jx}, \ \text{where} \ a_j \in \mathbb{R}. $$
I'm looking for simple, and hopefully elementary (in the spirit of elementary functions) examples of functions that cannot be represented by such a series.
My initial thought is that this space is equivalent to the space spanned by Taylor series (I haven't proved this, but it seems obvious), and so the complement of the space above is just the space of non-analytic functions. Any thoughts, clarifications or verifications on this would be much appreciated.