A point of discontinuity of a rational function is a point where the $x$-coordinate makes the function value undefined. Often, this occurs when the $x\text{-}$coordinate makes the denominator equal to $0.$ Below is an example.

Since $x=\text{-}1$ is not in the domain of the function $y=\frac{1}{x+1},$ it is a point of discontinuity. There is no way to rewrite the expression so that $x=\text{-}1$ becomes a valid input. Therefore, it is called a non-removable point of discontinuity. Below is another example. $$f(x)=\frac{x^2-4}{x-2}$$ In the function above, $x=2$ is a point of discontinuity because when $x$ equals $2,$ the denominator is equal to $0.$ However, the expression can be simplified using the differences of squares. After simplifying, the value $x=2$ is now a valid input. Therefore, it is called a removable point of discontinuity.

A removable discontinuity, like the one at $x=2,$ is typically called a hole in the graph.