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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 $P$ ?

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Becaue you can create a function which jumps different amounts and in different directions in each of the straight lines approaching your point. I don't think the idea is as simple. –  Alex Youcis Nov 30 '11 at 4:00
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@Alex Youcis : in this case (different amount of jumps in different directions)...for the function to be continuous in some neighborhood (except at that point) of the given point, do we need any restrictions on the amounts of jumps in different directions ? –  Rajesh D Nov 30 '11 at 4:25
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I would say "jump discontinuity" makes no sense for domain $\mathbb R^2$. –  GEdgar Nov 30 '11 at 4:39
    
I think this is a good question, but would be improved by asking if the concept of jump discontinuity can be extended to function from $\mathbb{R}^2$ to $\mathbb{R}$. –  Quinn Culver Nov 30 '11 at 4:42
    
@Quinn : I've edited –  Rajesh D Nov 30 '11 at 4:47
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2 Answers

up vote 2 down vote accepted

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 $\gamma(t)=(t,t^2)$, $g(\gamma(t))=0$.

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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.

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