Discontinuity at a domain point: removable, jump and essential cases, and the first- and second-kind grouping

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 $\epsilon$-$\delta$ 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 $A\cap (-\mathrm{\infty },c)$ and $A\cap (c,\mathrm{\infty })$ write

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

(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 $\epsilon$-$\delta$ condition of Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ 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 $\mathrm{\mid }x-c\mathrm{\mid }<\rho$ is $c$ itself and $\mathrm{\mid }f(c)-f(c)\mathrm{\mid }=0$. 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 $\epsilon$-$\delta$ 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 ${\lim }_{x\to c^{-}}f(x)\mathrm{\ne }{\lim }_{x\to c^{+}}f(x)$; 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 $\epsilon$-$\delta$ 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. At two-sided points, the first-kind/second-kind terminology groups the three cases above: the first kind includes both removable and jump discontinuities, and the second kind is essential. Thus the two kinds do not name the same three cases individually. At a one-sided point, the convention above uses only the available side: a first-kind discontinuity can also be removed by replacing $f(c)$ with that finite one-sided limit, but it is not a jump, since there are not two sides to compare. This endpoint extension is stated separately from the two-sided removable/jump split.