Types of Discontinuities

Not all functions are continuous. When a function fails to be continuous at a point, we classify the type of discontinuity. The three main types are: removable discontinuities (where a limit exists but doesn't match the function value), jump discontinuities (where left and right limits exist but differ), and essential discontinuities (where limits fail to exist). Understanding discontinuities is crucial for integration theory and Fourier analysis.

Definitions

Definition4.1One-sided limits

Let $f$ be defined on an interval containing $c$ (except possibly at $c$ itself). The right-hand limit of $f$ at $c$ is

$\lim_{x \to c^+} f(x) = L$

if for every $\epsilon > 0$ , there exists $\delta > 0$ such that $0 < x - c < \delta \implies |f(x) - L| < \epsilon$ .

Similarly, the left-hand limit is $\lim_{x \to c^-} f(x)$ .

RemarkTwo-sided limit

The (two-sided) limit $\lim_{x \to c} f(x) = L$ exists if and only if $\lim_{x \to c^+} f(x) = \lim_{x \to c^-} f(x) = L$ .

Types of discontinuities

Definition4.2Removable discontinuity

$f$ has a removable discontinuity at $c$ if $\lim_{x \to c} f(x)$ exists but either $f(c)$ is undefined or $\lim_{x \to c} f(x) \neq f(c)$ .

ExampleRemovable discontinuity

Let $f(x) = \frac{\sin(x)}{x}$ for $x \neq 0$ and $f(0) = 0$ . Then $\lim_{x \to 0} f(x) = 1 \neq f(0)$ , so $f$ has a removable discontinuity at $0$ . We can "remove" it by redefining $f(0) = 1$ .

Definition4.3Jump discontinuity

$f$ has a jump discontinuity at $c$ if $\lim_{x \to c^+} f(x)$ and $\lim_{x \to c^-} f(x)$ both exist but are not equal.

ExampleJump discontinuity

The Heaviside function

$H(x) = \begin{cases} 0 & x < 0, \\ 1 & x \geq 0 \end{cases}$

has a jump discontinuity at $x = 0$ : $\lim_{x \to 0^-} H(x) = 0$ and $\lim_{x \to 0^+} H(x) = 1$ . The jump size is $1 - 0 = 1$ .

Definition4.4Essential discontinuity

$f$ has an essential discontinuity (or infinite discontinuity) at $c$ if at least one of the one-sided limits fails to exist (or is infinite).

ExampleEssential discontinuity

Let $f(x) = \sin(1/x)$ for $x \neq 0$ . As $x \to 0$ , $1/x \to \pm\infty$ and $\sin(1/x)$ oscillates between $-1$ and $1$ infinitely often. Thus $\lim_{x \to 0} \sin(1/x)$ does not exist — $f$ has an essential discontinuity at $0$ .

ExampleInfinite discontinuity

$f(x) = 1/x$ at $x = 0$ has an essential discontinuity: $\lim_{x \to 0^+} f(x) = +\infty$ and $\lim_{x \to 0^-} f(x) = -\infty$ . Neither one-sided limit is finite.

Monotone functions and discontinuities

Theorem4.1Monotone functions have at most countably many discontinuities

If $f : [a, b] \to \mathbb{R}$ is monotone, then the set of discontinuities of $f$ is at most countable.

RemarkProof sketch

For a monotone increasing $f$ , each discontinuity is a jump discontinuity. At each jump, associate a rational number in the "jump interval." Since the rationals are countable, there are at most countably many jumps.

ExampleStaircase function

Define $f(x) = \lfloor x \rfloor$ (the greatest integer $\leq x$ ). Then $f$ is increasing and has jump discontinuities at every integer, with $\lim_{x \to n^-} f(x) = n - 1$ and $\lim_{x \to n^+} f(x) = n$ . There are countably many discontinuities.

Summary

Discontinuities are classified into three types:

Removable: limit exists, can be "fixed" by redefining the function.
Jump: one-sided limits exist but differ.
Essential: at least one limit fails to exist.
Monotone functions have at most countably many discontinuities (all jumps).

See Intermediate Value Theorem and Riemann Integration for how continuity and discontinuities affect analysis.