Singularity¶

Definition¶

A point $x_0$ is called a singularity of a function $f$ if $f$ fails to be defined or continuous at $x=x_0\text{,}$ while being defined and continuous in a punctured neighbourhood of $x_0\text{.}$

Lecture 5 classifies these problem points by their one-sided limits:

• Removable: the finite limit exists, but $f(x_0)$ is missing or has the wrong value.
• Jump: both one-sided limits are finite but unequal.
• Infinite: at least one one-sided limit is $+\infty$ or $-\infty\text{.}$
• Oscillatory: a one-sided limit fails because of persistent oscillation.

Interpretation¶

A singularity marks a point where a function fails to behave regularly, although it is defined and continuous at all points close to it. The question is what goes wrong at that single point, and the one-sided limits answer it.

The four types can be read as a sequence of questions. If both one-sided limits are finite and equal, the singularity is removable: defining or redefining $f(x_0)$ as the common limit makes $f$ continuous at $x_0$. If they are finite but different, no value of $f(x_0)$ helps, and the jump has size $f(x_0^+)-f(x_0^-)$$f(x_0^{-})$. If one of them is infinite, the graph has a vertical asymptote at $x=x_0$$x_0$. If a one-sided limit fails to exist because the values keep oscillating, the singularity is oscillatory.

A typical removable singularity: $\frac{\sin x}{x}$ is undefined at $0$ but has limit $1$ there, so setting the value $1$ at $0$ gives a function that is continuous on all of $\mathbb{R}$.

A common slip concerns points outside the domain. The function $1/x$ is continuous at every point of its domain $\mathbb{R}\setminus\{0\}$$\{0\}$; what fails at $0$ is that $1/x$ cannot be extended to a function continuous at $0$. Calling $0$ a discontinuity of $1/x$ should be read in this sense, not as a point of the domain where continuity fails.

The word returns later with improper integrals, where a function that is unbounded near a point, such as $1/\sqrt{x}$ near $0$, is integrated by taking a limit.

Key Results¶

• At an interior point, $f$ is continuous if and only if $f(x_0^-)=f(x_0^+)=f(x_0)\text{;}$ a discontinuity is exactly a failure of this equality.
• Several distinct things can go wrong at $x_0\text{:}$ $f(x_0)$ may be undefined while nearby values behave well; $f$ may be defined but not continuous; nearby values may become unbounded; or nearby values may oscillate without approaching anything.
• The word singularity returns in Lecture 12 for improper integrals: an integrand may be unbounded at an interior singularity, and the integral is then split there.

Examples¶

• The function $f(x) = (x^2 - 1)/(x - 1)$ has a removable discontinuity at $x=1\text{.}$
• The piecewise function $f(x)=x$ for $x\leq 0$ and $f(x)=x+1$ for $x>0$ has a jump discontinuity at $x=0\text{;}$ so does the signum function at the origin.
• The function $f(x)=1/x$ has an infinite discontinuity at $x=0\text{.}$
• The function $f(x) = \sin(1/x)$ has an oscillatory discontinuity at $x=0\text{.}$

The three graphs illustrate singularities at $x=0$$0$. The first is removable because changing the assigned value $-1$ to the limit $0$ restores continuity; the second has unequal one-sided limits $0.5$ and $1$; and the third, $1/x$, has an infinite singularity.

Prerequisites¶

• Continuity
• Domain

Related Concepts¶

• Limit at a Point
• Vertical Asymptote
• Zero of a Function

Sources¶

• Math I: Calculus · Lectures 5 and 12

Keywords: discontinuity, types of discontinuity, removable discontinuity, jump discontinuity Symbols: infinity, real numbers, square root

Last updated: 20 October 2026 03:02