So in the lecture notes I had, there were these definitions:
Let function $f$ be such that $f:U\to \mathbb{C}$ where $U$ is a domain,
Def 1: A point,$p$, is said to be a regular point if the function is holomorphic at $p$.
Def 2: A point, $q$, is said to be a singularity if it's a limit point of regular points but not holomorphic at $p$.
Hence if I have a function $f$ that is nowhere differentiable then are all the points in $U$ singularities or none the points are singularities, considering there are no regular points?
According to this definition no point is a singularity. In most books what is called a singularity here is called an isolated singularity.