Classification of discontinuities of multidimensional maps

Question

I know that for functions $f: \mathbb{R} \rightarrow \mathbb{R}$ there exist references which describe various discontinuities that these functions can exhibit. However, I could never find any source of information that would present a unified approach to the classification of the discontinuities of more general functions (for example, $f: \mathbb{R^n} \rightarrow \mathbb{R}$ or, at least $f: \mathbb{R^2} \rightarrow \mathbb{R}$). It would also be useful to know if there exists standard notation for some of the discontinuities that occur in such functions.