Hey, I need to numerically identify discontinuity points for a function given by a general expression (formula). I am able to evaluate the values at any point. I need it to be fast bu not accurate. The goal is to correctly render functions. With my naive algorithm, I get vertical lines on $x=0$ for $1/x$ and $sign(x)$. The types of discontinuities I need to find are like in the functions $sign(x)$ and $1/x$, that is -,+ adjacent polars and step like functions. I would like to avoid false positives like in the function $sin(1/x)$ which may numerically turn out as discontinues as you approach 0.

Thank You!!!