# 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 $f \left[{ I }\right]$ be everywhere dense in $J$, where $f \left[{ I }\right]$ denotes the image of $I$ under $f$.

Let $c \in I$.

Then:

$\displaystyle \left({ \lim_{ x \to c^{-} } f \left({ x }\right) \,.\,.\, \lim_{ x \to c^{+} } f \left({ x }\right) }\right) \cap f \left[{ I }\right] \subseteq \left\{ { f \left({ c }\right) } \right\}$

## Proof

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

Since $f$ is increasing:

$\displaystyle \lim_{ x \to c^{-} } f \left({ x }\right) < \lim_{ x \to c^{+} } f \left({ x }\right)$

Suppose $z \in \displaystyle \left({ \lim_{ x \to c^{-} } f \left({ x }\right) \,.\,.\, \lim_{ x \to c^{+} } f \left({ x }\right) }\right) \cap f \left[{ I }\right]$.

Then:

$\displaystyle \exists t \in I : f \left({ t }\right) \in \left({ \lim_{ x \to c^{-} } f \left({ x }\right) \,.\,.\, \lim_{ x \to c^{+} } f \left({ x }\right) }\right)$

### Case 1 : $t < c$

Suppose that $t < c$.

 $\displaystyle t$ $<$ $\displaystyle c$ $\displaystyle \implies \ \$ $\displaystyle f \left({ t }\right)$ $\le$ $\displaystyle \lim_{ x \to c^{-} } f \left({ x }\right)$ Definition of Increasing Real Function $\displaystyle \implies \ \$ $\displaystyle f \left({ t }\right)$ $\notin$ $\displaystyle \left({ \lim_{ x \to c^{-} } f \left({ x }\right) \,.\,.\, \lim_{ x \to c^{+} } f \left({ x }\right) }\right)$ Definition of Open Real Interval

So we may discard this case.

$\Box$

### Case 2 : $t = c$

Suppose that $t = c$.

 $\displaystyle t$ $=$ $\displaystyle c$ $\displaystyle \implies \ \$ $\displaystyle f \left({ t }\right)$ $=$ $\displaystyle f \left({ c }\right)$ Definition of Mapping $\displaystyle \implies \ \$ $\displaystyle f \left({ t }\right)$ $\in$ $\displaystyle \left({ \lim_{ x \to c^{-} } f \left({ x }\right) \,.\,.\, \lim_{ x \to c^{+} } f \left({ x }\right) }\right)$ Definition of Open Real Interval

So the proposition holds in this case.

$\Box$

### Case 3 : $c < t$

Suppose that $t > c$.

 $\displaystyle t$ $>$ $\displaystyle c$ $\displaystyle \implies \ \$ $\displaystyle f \left({ t }\right)$ $\ge$ $\displaystyle \lim_{ x \to c^{+} } f \left({ x }\right)$ Definition of Increasing Real Function $\displaystyle \implies \ \$ $\displaystyle f \left({ t }\right)$ $\notin$ $\displaystyle \left({ \lim_{ x \to c^{-} } f \left({ x }\right) \,.\,.\, \lim_{ x \to c^{+} } f \left({ x }\right) }\right)$ Definition of Open Real Interval

So we may discard this case.

$\Box$

So $f \left({ t }\right) = c$, by Proof by Cases.

Thus:

 $\displaystyle z$ $=$ $\displaystyle f \left({ t }\right)$ Definition of $t$ $\displaystyle$ $=$ $\displaystyle c$ From above $\displaystyle \implies \ \$ $\displaystyle z$ $\in$ $\displaystyle \left\{ { f \left({ c }\right) } \right\}$

Thus:

$\displaystyle z \in \left({ \lim_{x \to c^-} f \left({ x }\right) \,.\,.\, \lim_{ x \to c^+} f \left({ x }\right) }\right) \cap f \left[{ I }\right] \implies z \in \left\{ { f \left({ c }\right) } \right\}$

Hence the result, by definition of subset.

$\blacksquare$