TITLE: What is a continuous extension?
QUESTION [9 upvotes]: The continuous extension of $f(x)$ at $x=c$ makes the function continuous at that point.  

Can you elaborate some more? I wasn't able to find very much on "continuous extension" throughout the web.
How can you turn a point of discontinuity into a point of continuity? How is the function being "extended" into continuity?
Thank you.

REPLY [8 votes]: "Extending" a function $f$ means defining it at points where it wasn't defined before. As an example, the function
$$f(x):={\sin x\over x}\quad(x\in \dot{\mathbb R})$$
is not defined at $0$. But it makes very much sense to "extend" the domain of $f$ by defining $f(0):=1$. When it matters under the given circumstances, one could even denote the the extended function by $\tilde f:\ {\mathbb R}\to{\mathbb R}$.
When some $f$ a priori is defined on a subset $A\subset{\mathbb R}$, and if $\xi\in{\mathbb R}\setminus A$ is an accumulation point of $A$, then it makes sense to check whether the $$\lim_{x\to \xi,\ x\in A} f(x)$$
exists. Assume this is the case, so that $\lim_{x\to \xi,\ x\in A} f(x)=\eta$. Then defining
$$f(\xi):=\eta$$
extends $f$ to $\tilde A:=A\cup\{\xi\}$ and makes $f$ continuous at $\xi$.