I was doing some research and stumbled upon series of the form $$f(x)=\sum_{l=1}^{+ \infty}a_l \cdot (g(x))^{\frac {1}{l}}$$
A function $g(x)$ in my research is infinitely many times differentiable but $f(x)$ is discontinuos (although not wildly) and I am not sure can this happen at all?
That is, can a discontinuos $f(x)$ be developed into this kind of series at all if $g(x)$ is infinitely many times differentiable?
More precisely, $f$ has an infinite number of discontinuities and a set of discontinuities is nowhere dense.
Also, $g$ is strictly increasing function.