Surjective Monotone Function is Continuous

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

Let $Y$ be a real interval.

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

Then $f$ is continuous on $X$.

Proof
, let $f$ be increasing.

From Limit of Monotone Real Function: Corollary, the one sided limits of monotone function exist everywhere and satisfy:

These equations imply that:


 * $L^-_c$ and $ L^+_c$ are in $Y$

and:


 * $L^-_c \le L^+_c$.

Suppose that $\ds L = \lim_{x \mathop \to c} \map f x$ exists.

Then the left and right limits at $c$ are equal and the previous equations imply:


 * $L =L^-_c$

This leads to:
 * $L \ge \map f c$

Similarly: $L = L^+_c$

which leads to:
 * $L \le \map f c$

Hence:
 * $\ds \lim_{x \mathop \to c} \map f x = \map f c$

proving continuity at $c$.

By assumption, $f$ is increasing.

Suppose $\ds \lim_{x \mathop \to c} \map f x$ does not exist.

Then there is a jump discontinuity at $c$.

$f$ has a jump discontinuity at $c \in X$.

Then $y \in Y$ such that:


 * $L^-_c < y < L^+_c$

By surjectivity $y = \map f a$ for some $a \in X$.

Hence:


 * $L^-_c < \map f a < L^+_c$

If $a < c$ then $\map f a \le L^-_c$. This contradicts the previous inequality.

There is a similar contradiction if $a \ge c$.