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 $I^{-} = I \cap (-\infty,c)$ and $I^{+} = I \cap (c,\infty)$ 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 $A \cap (-\infty, c)$ and $A \cap (c, \infty)$). 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 $\varepsilon$-$\delta$ 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 $|(-f)(x) - (-f)(c)| = |f(x) - f(c)|$.

A point of $I$ that is not interior is an endpoint, and there are at most two. $I^{-} = \varnothing$ says that $c$ is a least element of $I$ and $I^{+} = \varnothing$ 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.