The three types of discontinuity

When continuity fails at a point, the discontinuity is classified into three types, according to how the one-sided limits behave.

In brief — Three types of discontinuity

1. First kind (jump): the one-sided limits exist and are finite but are different, $\lim_{x\to x_0^-}f(x) \ne \lim_{x\to x_0^+}f(x)$. The size of the jump is $\left|\lim^+ - \lim^-\right|$.
2. Second kind: at least one of the one-sided limits is infinite or does not exist ($\pm\infty$ or oscillating).
3. Third kind (removable): the limit exists and is finite, $\lim_{x\to x_0}f(x) = L$, but either $f(x_0)$ is not defined, or $f(x_0) \ne L$. The discontinuity can be “removed” by redefining $f(x_0) = L$.

The classification criterion concerns only the one-sided limits: if they exist finite but different one has a jump (first kind); if at least one is infinite or non-existent one has the second kind; if the limit exists but does not coincide with the value (or the value is missing) the discontinuity is removable (third kind).

The three types of discontinuity: jump (first kind), asymptote/infinite limit (second kind), removable hole (third kind).

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Topics: Continuita
Concepts: Discontinuita · Discontinuita prima specie · Discontinuita seconda specie · Discontinuita terza specie · Limite destro e sinistro
Methods: Continuita tipi
Skills: Interpretare grafico · Ragionare per casi