Question

What is a jump discontinuity of a graph?

Answer 1

A point in the graph of a function where left and right limits exist but differ.

Explanation:

Suppose `f(x)` is a real-valued function such that:

`{ (lim_(x->a^-) f(x) = u), (lim_(x->a^+) f(x) = v), (u != v) :}`

Then `f(x)` is said to have a jump discontinuity at `x=a`

For example, consider:

`f(x) = { (x + x/(abs(x)) " for " x != 0), (0 " for " x = 0) :}`

graph{x+x/abs(x) [-10, 10, -5, 5]}

This has a jump discontinuity at `x=0`, with:

`lim_(x->0^-) f(x) = -1`

`lim_(x->0^+) f(x) = 1`

Unlike a hole (a.k.a. removable discontinuity), there is no replacement value that we can assign to `f(x)` at a jump discontinuity in order to make `f(x)` continuous.