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:
- $\ds \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x} \cap f \sqbrk I \subseteq \set {\map f c}$
Proof
From Discontinuity of Monotonic Function is Jump Discontinuity, $\ds \lim_{x \mathop \to c^-} \map f x$ and $\ds \lim_{x \mathop \to c^+} \map f x$ are finite.
Since $f$ is increasing:
- $\ds \lim_{x \mathop \to c^-} \map f x < \lim_{x \mathop \to c^+} \map f x$
Suppose $z \in \ds \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x} \cap f \sqbrk I$.
Then:
- $\ds \exists t \in I : \map f t \in \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x}$
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:
- $\ds z \in \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x} \cap f \sqbrk I \implies z \in \set {\map f c}$
Hence the result, by definition of subset.
$\blacksquare$