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 \(\mathbb{Q}\).

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