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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.2011-11-30
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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 ?2011-11-30
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    I would say "jump discontinuity" makes no sense for domain $\mathbb R^2$.2011-11-30
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    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 $\mathbb{R}$.2011-11-30
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    @Quinn : I've edited2011-11-30
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    I guess that we may extend this concept by changing the point of view. If we attribute the existence of a jump continuity to the Dirac delta function term (or the corresponding singular measure) within the derivative of the function, we may extend this idea to functions on a plane, space or on a more general domain.2011-11-30
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    The question is: what properties of the jump discontinuity of $\mathbb{R}$ do you want the "extended" notion to have? (BTW: your "edit" equally well applies to a point-wise blow-up like $1/x$ or to removable discontinuities, so is not very useful.)2011-11-30
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    Something like $z=\frac{y}{\sqrt{x^2+y^2}}$, perhaps? (See WolframAlpha: http://www.wolframalpha.com/input/?i=y%2Fsqrt%28x%5E2%2By%5E2%29) We have continuity everywhere except the origin, and the limiting $z$ from approaching the origin along the line $y=mx$ is $m/\sqrt{1+m^2}$.2011-11-30
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    @DayLateDon: why not expand that into an answer? Graphics get attention, if you can supply them. :-)2012-04-06

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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.