# What's the difference between jump and removable discontinuity?

May 7, 2015

First of all, we are talking about a behavior of a function around some particular value of an argument.

Let's assume our function is represented as $y = f \left(x\right)$ and its behavior around the value of argument $x = a$ is what we are analyzing.

Also assume that the following two limits are properly defined, exist and are finite:
1. Limit of the function as argument $x$ approaches a value $a$ from the left:
$c = {\lim}_{x \to {a}^{-}} f \left(x\right)$
2. Limit of the function as argument $x$ approaches a value $a$ from the right:
$d = {\lim}_{x \to {a}^{+}} f \left(x\right)$

If the above limits are equal and a function $f \left(x\right)$ is defined at $x = a$ and its value is the same as these limits, we have no discontinuity.
In other words, the condition for not having a discontinuity is
${\lim}_{x \to {a}^{-}} f \left(x\right) = {\lim}_{x \to {a}^{+}} f \left(x\right) = f \left(a\right)$
Example of a function with no discontinuity is
$y = | x |$:
graph{|x| [-10, 10, -5, 5]}

If the above limits are not equal, we have a jump discontinuity.
In other words, the condition for a jump discontinuity is
${\lim}_{x \to {a}^{-}} f \left(x\right) \ne {\lim}_{x \to {a}^{+}} f \left(x\right)$
Example of a function with a jump discontinuity at $x = 0$ is
$y = - 1$ for negative $x$,
$y = 1$ for positive $x$ and
$y = 0$ for $x = 0$:
graph{x/|x| [-10, 10, -5, 5]}

Finally, If the above limits are equal but a function $f \left(x\right)$ is either not defined at $x = a$ or is defined, but its value is not the same as these limits, we have a removable discontinuity.
In other words, the condition for a removable discontinuity is
${\lim}_{x \to {a}^{-}} f \left(x\right) = {\lim}_{x \to {a}^{+}} f \left(x\right)$ AND
$f \left(a\right)$ IS UNDEFINED OR $f \left(a\right) \ne {\lim}_{x \to {a}^{-}} f \left(x\right)$
Example of a function with removable discontinuity is
$y = {x}^{2}$ everywhere except $x = 0$ and
function is undefined at $x = 0$ :
graph{x^3/x [-10, 10, -5, 5]}