A monotone function on an interval has no discontinuity of the second kind: at every point both relevant one-sided limits exist, and an interior point $c$ is a discontinuity exactly when ${\lim }_{x\to c^{-}}f(x)<{\lim }_{x\to c^{+}}f(x)$

Let $I\subseteq \mathbb{R}$ be order-convex (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $f:I\to \mathbb{R}$ be nondecreasing (Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences). Write $I^{-}=I\cap (-\mathrm{\infty },c)$ and $I^{+}=I\cap (c,\mathrm{\infty })$ for $c\in I$.

1. At every $c\in I$, each of the two one-sided limits that is well posed exists (The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A\cap (-\mathrm{\infty },c)$ and $A\cap (c,\mathrm{\infty })$). Consequently $f$ has no discontinuity of the second kind (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): every discontinuity of $f$ is of the first kind.
2. Call $c\in I$ an interior point of $I$ when both $I^{-}$ and $I^{+}$ are nonempty. At such a point ${\lim }_{x\to c^{-}}f(x)\le f(c)\le {\lim }_{x\to c^{+}}f(x),$ and $f$ is 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) if and only if ${\lim }_{x\to c^{-}}f(x)={\lim }_{x\to c^{+}}f(x)$.
3. Hence an interior point $c$ is a discontinuity of $f$ exactly when ${\lim }_{x\to c^{-}}f(x)<{\lim }_{x\to c^{+}}f(x),$ and every such discontinuity is a jump, of jump ${\lim }_{x\to c^{+}}f(x)-{\lim }_{x\to c^{-}}f(x)>0$.

The same three claims hold for a nonincreasing $f$, with the two one-sided limits exchanged and all inequalities reversed, by applying the above to $-f$, which is nondecreasing (Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences) and has exactly the same points of continuity, since $\mathrm{\mid }(-f)(x)-(-f)(c)\mathrm{\mid }=\mathrm{\mid }f(x)-f(c)\mathrm{\mid }$.

A point of $I$ that is not interior is an endpoint, and there are at most two. $I^{-}=\mathrm{\varnothing }$ says that $c$ is a least element of $I$ and $I^{+}=\mathrm{\varnothing }$ that it is a greatest one, and a set has at most one of each. Those two points are excluded from claims 2 and 3 only because a comparison of two one-sided limits is not available there; claim 1 covers them.

• Nothing here counts the discontinuities. Claim 3 says only what a discontinuity of a monotone function looks like at an interior point. That the set of them is at most countable is a further theorem, Froda's theorem: the set of discontinuities of a monotone function on an interval is at most countable, the injection into $\mathbb{N}$ being built from one fixed enumeration of the rationals by least index, so no choice principle is used, and its proof is exactly the observation that the open intervals $({\lim }_{x\to c^{-}}f(x),{\lim }_{x\to c^{+}}f(x))$ attached to distinct discontinuities are disjoint.

• Why no interior discontinuity of a monotone function is removable. Claim 2 rules them out at interior points: there $L^{-}=L^{+}$ already forces continuity, because the inequality $L^{-}\le f(c)\le L^{+}$ pins $f(c)$ between the two one-sided values. That inequality is special to monotone functions, and it is what makes jump the only kind of interior discontinuity available. At a point of $I$ that is not interior the inequality is one-sided too and the argument does not apply, so a monotone function may fail to be continuous at an endpoint of $I$ while having its one one-sided limit; that failure is a discontinuity of the first kind and it is not a jump, there being only one side to compare.