I went over a few post on this forum and people cited a page on Mathworld saying that discontuity of a $\mathbb R^n$ function is hard to define.
However, that Mathworld page does implicitly defined a type directional limit and a type of essential discontinuity:
However, that page does not rigorously define what is directional limit.
**Question: ** what is a proper definition for directional limit and essential discontinuity?
I try to define my own:
Say the limit at zero via directional vector $v$ is $\lim_{x\to > 0+\epsilon v}f(x)$.
I also try to define essential discontinuity:
$f$ is strictly essentially discontinuous at $0$ if the directional limit to zero does not exists (and not infinite) for all directional vector $v$.
$f$ is weakly essentially discontinuous at $0$ if the directional limit to zero does not exists (and not infinite) for one direction directional vector $v$.
I also learn that a directional limit might be more complicated that the directional limit via a vector. For example, there are things called "radical limit", "nontangential limit", "stolz angle limit".