What is jump discontinuity in math?

A jump discontinuity of a graph occurs when the function has real, but different, right-hand and left-hand limits for the same x-value.

Look at the image below.

The function ##h(x)=(|x|)/x## has a jump discontinuity at x = 0.

Notice that ##lim_(x->0^+)=1## and ##lim_(x->0^-)=-1##

Here's another example. Let

This is what's known as a piece-wise defined function because we define it differently for different sub-intervals of its domain.

Notice that in this case

##lim_(x->1^-)f(x)=-3## and ##lim_(x->1^+)f(x)=1.5##

Therefore, the graph will have a jump discontinuity at x = 1. See image below.