Monotone Real Function with Everywhere Dense Image is Continuous/Lemma

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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.

$\blacksquare$