# Monotone Real Function with Everywhere Dense Image is Continuous/Lemma

## Theorem

Let $I$ and $J$ be real intervals.

Let $f: I \to J$ be a monotone real function.

Let $f \sqbrk I$ denote the image of $I$ under $f$.

Let $f \sqbrk I$ be everywhere dense in $J$.

Let $c \in I$.

Then:

$\displaystyle \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c+ } \map f x} \cap f \sqbrk I \subseteq \set {\map f c}$

## Proof

From Discontinuity of Monotonic Function is Jump Discontinuity, $\displaystyle \lim_{x \mathop \to c^-} \map f x$ and $\displaystyle \lim_{x \mathop \to c^+} \map f x$ are finite.

Since $f$ is increasing:

$\displaystyle \lim_{x \mathop \to c^-} \map f x < \lim_{x \mathop \to c^+} \map f x$

Suppose $z \in \displaystyle \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x} \cap f \sqbrk I$.

Then:

$\displaystyle \exists t \in I : \map f t \in \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x}$

### Case 1 : $t < c$

Suppose that $t < c$.

 $\ds t$ $<$ $\ds c$ $\ds \leadsto \ \$ $\ds \map f t$ $\le$ $\ds \lim_{x \mathop \to c^-} \map f x$ Definition of Increasing Real Function $\ds \leadsto \ \$ $\ds \map f t$ $\notin$ $\ds \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x}$ Definition of Open Real Interval

So we may discard this case.

$\Box$

### Case 2 : $t = c$

Suppose that $t = c$.

 $\ds t$ $=$ $\ds c$ $\ds \leadsto \ \$ $\ds \map f t$ $=$ $\ds \map f c$ Definition of Mapping $\ds \leadsto \ \$ $\ds \map f t$ $\in$ $\ds \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x}$ Definition of Open Real Interval

So the proposition holds in this case.

$\Box$

### Case 3 : $c < t$

Suppose that $t > c$.

 $\ds t$ $>$ $\ds c$ $\ds \leadsto \ \$ $\ds \map f t$ $\ge$ $\ds \lim_{x \mathop \to c^+} \map f x$ Definition of Increasing Real Function $\ds \leadsto \ \$ $\ds \map f t$ $\notin$ $\ds \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x}$ Definition of Open Real Interval

So we may discard this case.

$\Box$

So $\map f t = c$, by Proof by Cases.

Thus:

 $\ds z$ $=$ $\ds \map f t$ Definition of $t$ $\ds$ $=$ $\ds c$ from above $\ds \leadsto \ \$ $\ds z$ $\in$ $\ds \set {\map f c}$

Thus:

$\displaystyle z \in \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x} \cap f \sqbrk I \implies z \in \set {\map f c}$

Hence the result, by definition of subset.

$\blacksquare$