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

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*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.

*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 $\ell \le f(c) \le m$."

*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.

