TITLE: Example of a function continuous at only one point.
QUESTION [12 upvotes]: Possible Duplicate:
Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point? 

I want to know some example of a continuous function which is continuous at exactly one point.
We know that $f(x)=\frac{1}{x}$ is continuous everywhere except at $x=0$. But i think this in reverse manner but i dont get any example. So please help me out!

REPLY [31 votes]: One standard example is the function 
$$f(x)=\begin{cases}
x,&\text{if }x\in\Bbb Q\\
0,&\text{if }x\in\Bbb R\setminus\Bbb Q\;.
\end{cases}$$
That is, $f(x)=x$ if $x$ is rational, and $f(x)=0$ if $x$ is irrational. This function is continuous only at $x=0$.
Added: The same basic idea can be used to build a function that is continuous at any single specified point. With a little more ingenuity, you can use it to get, for instance, a function that is continuous just at the integers:
$$f(x)=\begin{cases}
\sin\pi x,&\text{if }x\in\Bbb Q\\
0,&\text{if }x\in\Bbb R\setminus\Bbb Q\;.
\end{cases}$$
This works because $\sin\pi x=0$ if and only if $x\in\Bbb Z$.

REPLY [12 votes]: Just take something like the Dirichlet function:
$$f : \mathbb R \ni x \mapsto \begin{cases} x&\text{if}\; x\in \mathbb Q\\0&\text{otherwise}\end{cases}$$
Then $f$ is continuous only at $x=0$.