Discontinuity of Monotonic Function is Jump Discontinuity

Theorem
Let $X$ be an open subset of $\R$.

Let $f: X \to Y$ be a monotone real function.

Then $f$ is discontinuous at a point $c \in X$ $c$ is a jump discontinuity of $f$.

Proof
The backwards implication follows directly from definition of a jump discontinuity.

For the forwards implication, we prove the contrapositive.

Let $X$ be an open subset of $\R$.

Let $f: X \to Y$ be a monotone real function.

suppose $f$ is increasing.

If $f$ is decreasing, note that $-f$ is increasing and we can simply replace $f$ by $-f$ in the following analysis.

Suppose that $c \in X$ is not a jump discontinuity of $f$.

By definition of jump discontinuity, it is not the case that:
 * $\ds \lim_{x \mathop \to c^-} \map f x$ and $\ds \lim_{x \mathop \to c^+} \map f x$ exist and are not equal.

By definition of an open set:
 * there exists $\epsilon_0 \in \mathbb R$ such that $\openint {c - \epsilon_0} {c + \epsilon_0} \subseteq X$.

By Limit of Monotone Real Function/Increasing/Corollary, since $c \in \openint {c - \epsilon_0} {c + \epsilon_0}$:
 * $\map f {c^-}$ and $\map f {c^+}$ both exist;
 * $\map f {c^-} \le \map f c \le \map f {c^+}$

Hence by Modus Ponendo Tollens:
 * $\map f {c^-} = \map f {c^+}$

Therefore we have:
 * $\map f {c^-} = \map f c = \map f {c^+}$

By Limit iff Limits from Left and Right:
 * $\map f x \to \map f c$ as $x \to c$

Hence $f$ is continuous at $c$ by definition.

Thus we have shown the contrapositive of the forwards implication.