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 $L = \lim_{x \to a} f(x)$ 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 $0$.

Explanations

L1 human

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