> [!definition] > > $ > f^\prime(x) = \begin{cases*} > f(x) &$f(x)$ exists \\ > \lim_{c \to x}f(c) &otherwise > \end{cases*} > $ > Let $f: A \to M$ be a [[Function|function]] and $c$ be a [[Cluster Point|cluster point]] of $A$, but $c \not\in A$ ($f(c)$ is not defined). If the [[Limit|limit]] of the function at that point exists and is $L$, then the function can be **extended** such that it is [[Continuity|continuous]] at $c$, by filling in $f(c)$ as the value of the limit. > > If the limit does not exist, then the function cannot be continuously extended. > [!theorem] > > Let $(a, b)$ be a bounded, open interval and $f: (a, b) \to \real$ be a continuous function, then the following are equivalent: > - $f$ is uniformly continuous in $(a, b)$. > - $f$ can be continuously extended to $[a, b]$. > > *Proof*. If $f$ can be continuously extended, then the extended version is uniformly continuous (being in a compact interval), which also works for $(a, b)$. > > Let $(x_n)$ be a [[Sequence|sequence]] in $(a, b)$ that converges to $a$, then $(x_n)$ is [[Cauchy Sequence|Cauchy]], and $f(x_n)$ is also Cauchy. Therefore $\exists \lim_{x \to a}f(x)$, and $f(x)$ can be extended to $a$.