Discontinuity of $f$ at a point of its domain, and its classification: removable discontinuity, jump discontinuity and essential discontinuity, equivalently Rudin's discontinuities of the first and of the second kind

Let $A \subseteq \mathbb{R}$, let $f : A \to \mathbb{R}$ and let $c \in A$. Then $f$ is discontinuous at $c$, and $c$ is a discontinuity of $f$, when $f$ is not continuous at $c$ (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point). As in The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$ write

$A^{-} := A \cap (-\infty, c), \qquad A^{+} := A \cap (c, \infty)$

(Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and recall that $\lim_{x \to c^{-}} f(x)$ is defined only when $c$ is a limit point of $A^{-}$, and $\lim_{x \to c^{+}} f(x)$ only when $c$ is a limit point of $A^{+}$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$).

At an isolated point there is nothing to classify. If $c$ is an isolated point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$), so that $A \cap N_{\rho}(c) = \{c\}$ for some real $\rho > 0$, then $f$ is continuous at $c$: the $\varepsilon$-$\delta$ condition of Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point is satisfied by $\delta := \rho$, since the only $x \in A$ with $|x - c| < \rho$ is $c$ itself and $|f(c) - f(c)| = 0$. So every discontinuity is a limit point of $A$, and the classification below covers every case that occurs.

Two-sided points

Suppose $c$ is a limit point of both $A^{-}$ and $A^{+}$, so that both one-sided limits are well posed. Say that $c$ is a discontinuity

• of the first kind when both one-sided limits exist;
• of the second kind, also called essential, when at least one of the two one-sided limits fails to exist.

A discontinuity of the first kind is further

• removable when $\lim_{x \to c^{-}} f(x) = \lim_{x \to c^{+}} f(x)$; the common value is then different from $f(c)$, for otherwise If $c$ is a limit point of the domain from both sides, the limit exists iff both one-sided limits exist and agree would give $\lim_{x \to c} f(x) = f(c)$ and $f$ would be continuous at $c$ (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point);
• a jump when $\lim_{x \to c^{-}} f(x) \ne \lim_{x \to c^{+}} f(x)$; the difference $\lim_{x \to c^{+}} f(x) - \lim_{x \to c^{-}} f(x)$ is then called the jump of $f$ at $c$.

The three cases removable, jump, essential are mutually exclusive and exhaust the two-sided discontinuities of $f$: either both one-sided limits exist, and then they are equal or not, or one of them does not exist.

Removable is a name for what can be repaired. If $c$ is a removable discontinuity with common one-sided value $L$, then the function agreeing with $f$ off $c$ and taking the value $L$ at $c$ is continuous at $c$, again by If $c$ is a limit point of the domain from both sides, the limit exists iff both one-sided limits exist and agree and Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point: changing the single value $f(c)$ removes the discontinuity. No such repair is available at a jump or at an essential discontinuity, since there the two-sided limit does not exist at all and no choice of value at $c$ can create it.

One-sided points

If $c$ is a limit point of exactly one of $A^{-}$ and $A^{+}$, only that side is defined and only that side is used: $c$ is a discontinuity of the first kind when the one-sided limit on the side in question exists, and of the second kind otherwise. When it exists it is different from $f(c)$, since on such a point the one-sided condition and the continuity condition are the same condition; and there is no jump case, there being nothing to compare the value with. The endpoints of an interval are the typical instance.

On the two vocabularies. First kind and second kind are Rudin's terms and are recorded because the literature uses them; removable, jump and essential are the names used in the rest of this library. They name the same three cases and no third classification is introduced.