Prove a monotone function at all points of its domain is either continuous or has a jump discontinuity

Question

Prove that at all points in its domain, a monotone function mapping an open set to $\mathbb R$ is either continuous or has a jump discontinuity. (A jump discontinuity is a point where $\lim^- \neq \lim^+$, as opposed to removable discontinuities or essential discontinuities.)

Note: Proofs are available. This question is to verify, critique, and improve my proof and its exposition.

Lemma: Let $A$ be an open subset of $\mathbb R$ and let $f: A \to \mathbb R$ be an increasing function. Then for all $c \in A$, $\lim_{x \to c^-} f(x)$ and $\lim_{x \to c^+} f(x)$ are defined, and $\lim_{x \to c^-} f(x) \leq f(c) \leq \lim_{x \to c^+} f(x)$.

Proof: Let $\ell = \sup \{f(x) : x \in A, x < c\}$. This set is non-empty and bounded by $f(c)$, and hence has a supremum. For any $\varepsilon > 0$, there exists an $a \in A, a < c$ where $\ell - \varepsilon < f(a) \leq \ell$, for if there were not, then $\ell - \varepsilon$ would be an upper bound. Since $f$ is increasing, for all $x$ such that $a < x < c$ we have $\ell - \varepsilon < f(a) \leq f(x) \leq \ell$, so $\ell = \lim_{x \to c^-} f(x)$. For a similar reason, $f(c) \geq \ell$.

Main Proof: For $x \in A$, we have $\ell := \lim_{x \to c^-} f(x)$ by the lemma and $m := \lim_{x \to c^+} f(x)$ by a similar argument. Suppose that $\ell = m$, and recall that if $\lim^- = \lim^+$, then the limit is defined and equal to $\lim^-$. Then since $\ell \leq f(c) \leq m$, $f(x) = \ell$, and the function is equal to its limit, and therefore continuous, at $x$.

If, alternatively, $\ell \neq m$, we have by definition a jump discontinuity at $x$.

Discussion: Is this proof correct and rigorous? Well-written? How can it be improved?

Answer 1

This proof is basically correct and also this is a good approach to proving this result. A reasonable mathematical reader would be able to tell what you mean already, but you could still clean up wording/typos a little bit:

1. Your Lemma says that it will prove things about $\lim_{x \to c^-}$ and also $\lim_{x \to c^+}$ but then you only talk about $\lim_{x \to c^-}$. You could add a sentence like "The claim about $\lim_{x \to c^+}$ follows by the same argument but using $\inf\{f(x) : x \in A, x > c\}$ instead of the supremum we used here." Any reasonable mathematical reader would already understand that this is what you meant, but still good to mention it.
2. In the Main Proof, you wrote: "we have $\ell := \lim_{x \to c^-} f(x)$ by the lemma and $m := \lim_{x \to c^+} f(x)$ by a similar argument." The second part is not by a "similar argument" though, they are just both directly applying the claims of the Lemma. I would just write: "By the Lemma, we know $\ell := \lim_{x \to c^-} f(x)$ and $m := \lim_{x \to c^+} f(x)$ exist and $\ell \le f(c) \le m$."
3. Near the end of your Main Proof, there's a sentence that starts with "Then since...". This sentence mentions $x$ twice but I think you meant $c$ both times.