Monotone functions have every relevant finite one-sided limit; interior discontinuities are jumps, with side limits ordered by the direction of monotonicity

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 at a domain point: removable, jump and essential cases, and the first- and second-kind grouping): 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 ${\lim }_{x\to c^{+}}f(x)-{\lim }_{x\to c^{-}}f(x)>0$.

For a nonincreasing $f$, every well-posed one-sided limit likewise exists, and at an interior point the side labels stay fixed while the order reverses: ${\lim }_{x\to c^{-}}f(x)\ge f(c)\ge {\lim }_{x\to c^{+}}f(x)$. Continuity is still equivalent to equality of the two limits. An interior discontinuity is therefore a jump exactly when ${\lim }_{x\to c^{-}}f(x)>{\lim }_{x\to c^{+}}f(x)$, with signed jump ${\lim }_{x\to c^{+}}f(x)-{\lim }_{x\to c^{-}}f(x)<0$. Apply the nondecreasing claims to $-f$: negation preserves the side of a limit and negates its value, and preserves continuity because $\mathrm{\mid }(-f)(x)-(-f)(c)\mathrm{\mid }=\mathrm{\mid }f(x)-f(c)\mathrm{\mid }$ (Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences, The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A\cap (-\mathrm{\infty },c)$ and $A\cap (c,\mathrm{\infty })$).

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 in the nondecreasing case 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.