We know that for a function $f \colon \mathbb{R} \to \mathbb{R}$, a jump discontinuity at a point $P$ is defined as the left and right limits exist but not equal. I'd like to know if this concept can be extended to functions of the form $f\colon\mathbb{R}^2 \to \mathbb{R}$, as here there is no such concept as left and right limits.
EDIT : my idea is to keep the requirement that there need to be a neighborhood of the point $P$, where $f$ is continuous except at $P$.
EDIT 2 : In addition to the above condition that $f$ is continuous in some neighborhood of $P$ except at $P$, as suggested by Alex Youcis (in comments) it can be proposed that there be different amounts of jump along different tangent vectors (different directions) at $P$, but do wee need any condition on the amounts of jump in order for $f$ to satisfy condition 1, i.e., $f$ being continuous in some neighborhood of $P$ except at $P$ ?