Examples of classification

Let us look at two concrete cases: a jump (first kind) and a removable discontinuity (third kind).

Example — First-kind discontinuity (jump)

Let $f(x) = \begin{cases} x^2 & x<1 \\ 3 & x=1 \\ 2x & x>1\end{cases}$ Then $\lim_{x\to 1^-}x^2 = 1$; $\lim_{x\to 1^+}2x = 2$; $f(1) = 3$.

The one-sided limits are finite but different ($1\ne 2$): this is a first-kind discontinuity (jump of size $|2-1|=1$). The value $f(1)=3$ is irrelevant for the classification: what matters are the limits.

Example — Removable discontinuity (third kind)

Let $f(x) = \dfrac{\sin x}{x}$ for $x\ne 0$, undefined at $x=0$. By the standard limit $\lim_{x\to 0}\frac{\sin x}{x} = 1,$ the limit exists and is finite, but the point $x=0$ does not belong to the domain: the discontinuity is removable (third kind). It is enough to set $f(0)=1$ to make the function continuous.

The comparison is instructive: in the first case the limit at $x_0$ does not exist (the two one-sided limits differ), so the discontinuity is irreparable; in the second the limit exists, and precisely for this reason the function can be “repaired” by redefining it at the point.

Links

Topics: Continuita
Concepts: Discontinuita prima specie · Discontinuita terza specie · Limite notevole
Skills: Calcolare limiti · Ragionare per casi