Can the concept of a  jump discontinuity be extended to functions of the form $f\colon \mathbb{R}^2 \to \mathbb{R}$ 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 $P$ ?
 A: What you are looking for are probably functions of bounded variation. These functions are a slight but powerful generalization of weakly differentiable functions with integrable weak derivative (i.e. of the Sobolev space $W^{1,1}$). Functions of bounded variation do not have to be continuous but still have a lot of structure, e.g. the first distributional derivative can be interpreted as a regular vector measure and there is a notion of curvature of the level sets.
Especially, the "jump set" of functions of bounded variation obeys certain restrictions and it is true that they only have jump type discontinuities .
As a reference I suggest the book "Measure theory and fine properties of functions" by Evans, Gariepy.
A: One generalization of the one-dimensional function with a jump-discontinuity might be something like
$$
f(x,y) = \frac{x}{\sqrt{x^2+y^2}}
$$
Along each line through the origin, except the line $x=0$, there is a jump discontinuity of size $\dfrac{2}{\sqrt{1+m^2}}$, where the equation of the line is $y=mx$.
However, the situation is more complex in $\mathbb{R}^2$. For example, the limit of the function
$$
g(x,y)=\frac{x^2-y}{(x^2+y^2)^2+x^2-y}
$$
along each ray terminating at the origin is $1$. However, the function is not continuous at the origin since along the curve $\gamma(t)=(t,t^2)$, $g(\gamma(t))=0$.
