Removable discontinuity

Builds on Limit of a function Continuity at a point

Definition

Let $D \subseteq \mathbb{R}$ , let $f \colon D \to \mathbb{R}$ , and let $a \in D$ be a limit point of $D$ . The point $a$ is a removable discontinuity of $f$ if the limit $L = \lim_{x \to a} f(x)$ exists and $L \ne f(a)$ .

Remarks

Change the value of $f$ at $a$ to $L$ . The new function is continuous at $a$ . For example, let $f(x) = 1$ for $x \ne 0$ and $f(0) = 0$ . Then $0$ is a removable discontinuity. Some texts also call $a$ removable when $a \notin D$ and the limit exists. An example is $\sin(x)/x$ at $0$ .

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`0QG9C579SR`
URI	`m2://0QG9C579SR`
Kind	definition
Writing	W1 stated
MSC	26A15
Defines	Removable discontinuity
Uses	Limit of a function, Continuity at a point

Removable discontinuity

Definition

Remarks

Reading path

Further reading

What this page needs for W2

What links here