# Real Analysis

Discontinuous Functions

# Discontinuity

Consider $$f : (a, b) \rightarrow \mathbb{R}$$, which is discontinuous at $$x$$ with $$f(x) = q$$ and is continuous everywhere else. Write $$f(x^+) \rightarrow q$$ or $$\lim\limits_{p \rightarrow x^+} f(p) = q$$. Similarly, $$f(x^-) \rightarrow q'$$ or $$\lim\limits_{p \rightarrow x^-} f(p) = q'$$.

$$\lim\limits_{p \rightarrow x} f(p)$$ exists if and only if $$\lim\limits_{p \rightarrow x^+} f(p) = \lim\limits_{p \rightarrow x^-} f(p)$$. If $$f$$ is discontinuous at $$x$$ but $$\lim\limits_{p \rightarrow x^+} f(p)$$ and $$\lim\limits_{p \rightarrow x^-} f(p)$$ exist, say $$f$$ has a discontinuity of the first kind (or a simple discontinuity). Otherwise, say $$f$$ has a discontinuity of the second kind.

For example,

1. Dirichlet function: \begin{align} f(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \\ 0 & \text{otherwise} \end{cases} \end{align} $$f$$ has discontinuities of the second kind at every point.
2. \begin{align} f(x) = \begin{cases} \frac{1}{q} & \text{if } x = \frac{p}{q} \text{ in lowest terms} \\ 0 & \text{otherwise} \end{cases} \end{align} $$f$$ has discontinuities of the first kind at rationals (all discontinuities are simple) but is continuous at irrationals.
3. \begin{align} f(x) = \begin{cases} 0 & \text{if } x \leq 0 \\ \sin\left(\frac{1}{x}\right) & \text{otherwise} \end{cases} \end{align} $$f$$ has a discontinuity of the second kind at $$0$$.
4. \begin{align} f(x) = \begin{cases} x^2 & \text{if } x \in \mathbb{Q} \\ 0 & \text{otherwise} \end{cases} \end{align} $$f$$ is continuous at $$0$$ and all the discontinuities are of the second kind.

# Monotonic Functions

A function $$f : (a, b) \rightarrow \mathbb{R}$$ is monotonically increasing if $$x \leq y$$ implies $$f(x) \leq f(y)$$. A function $$f : (a, b) \rightarrow \mathbb{R}$$ is monotonically decreasing if $$x \leq y$$ implies $$f(x) \geq f(y)$$.

Theorem. If $$f : (a, b) \rightarrow \mathbb{R}$$ is monotonically increasing then $$\sup\limits_{t \in (a, x)} f(t) \leq f(x) \leq \inf\limits_{t \in (x, b)} f(t)$$.

Theorem. If $$f : (a, b) \rightarrow \mathbb{R}$$ is monotonically increasing/decreasing on $$(a, b)$$ then $$f(x^+)$$ and $$f(x^-)$$ exist for all $$x \in (a, b)$$.

Corollary. Monotonic functions have no discontinuity of the second kind.

Theorem. If $$f : (a, b) \rightarrow \mathbb{R}$$ is monotone on $$(a, b)$$ then the set of points where $$f$$ is not continuous is countable.

Proof idea: let $$D$$ be the set of points where $$f$$ is discontinous. For every point $$x \in D$$, pick $$r(x) \in \mathbb{Q}$$ such that $$f(x^-) < r(x) < f(x^+)$$. If $$x, y \in D$$ then $$r(x) \neq r(y)$$ because $$f$$ is monotone. Get a 1-1 correspondence between $$D$$ and a subset of $$\mathbb{Q}$$.