---
tag: 0QG9C579SR
kind: definition
name: Removable discontinuity
msc: [26A15]
defines: 0K96Z2S699
uses: [0TXTSEMHR5, 0EEE2CZB2A]
sources:
  - url: https://en.wikipedia.org/wiki/Classification_of_discontinuities
    locator: Removable discontinuity
---

## 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|0TXTSEMHR5]] $L = \lim_{x \to a} f(x)$ exists and $L \ne f(a)$.

## Remarks

Change the value of $f$ at $a$ to $L$. The new function is [[continuous at|0EEE2CZB2A]] $a$.
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 $0$.

## Explanations

### L1 human

At a removable discontinuity, the graph has a single misplaced point.
Moving this one point repairs the continuity.