Jump discontinuity is a non removable discontinuity that occurs when a function f(x) is defined at x = a, but $\lim_{x\rightarrow a}f(x)$ does not exist.
This phenomena can be observed in the floor function f(x) = $\left \lfloor x \right \rfloor$ (Greatest integer function) and ceiling function
f(x) = $\left \lceil x \right \rceil$ (Least integer function) at every integer value of x.



The right limit does not exist at integer values of x for the floor function, which causes the jump discontinuities.


The left limit does not exist at integer values of x for the ceiling function, which causes the jump discontinuities.

Jump discontinuity also occurs in some piecewise defined functions.
Consider the piecewise defined function,
f(x) = $\left\{\begin{matrix}
\frac{1}{2}x^{2} & if & x<2\\
x+1 & if & x\geq 2
\end{matrix}\right.$
The function is defined at x = 2 on the straight line y = x + 1 i.e. f(2) = 3.
$\lim_{x\rightarrow 2^{}}$ =
$\frac{1}{2}$ $\times$ 4 = 2 and $\lim_{x\rightarrow 2^{+}}$ = 3
Since the left and right limits are not equal, the limit does not exist for the function.
This has caused the jump discontinuity.