Types of Discontinuities

There are several ways that a function can fail to be continuous. The three most common are:

If $\displaystyle{\lim_{x \to a^+} f(x)}$ and $\displaystyle{ \lim_{x \to a^-} f(x)}$ both exist, but are different, then we have a jump discontinuity. If either $\displaystyle{\lim_{x \to a^+} f(x)} = \pm \infty$ or $\displaystyle{\lim_{x \to a^-} f(x)} = \pm \infty$, then we have an infinite discontinuity, also called an asymptotic discontinuity. If $\displaystyle{\lim_{x \to a^+} f(x)}$ and $\displaystyle{\lim_{x \to a^-} f(x)}$ exist and are equal (and finite), but $f(a)$ happens to be different (or doesn't exist), then we have a removable discontinuity, since by changing the value of $f(x)$ at a single point we can make $f(x)$ continuous.

These kinds of discontinuities are explained, with examples, in the following video: