Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point? Is the function $f(x) = \lim_{k \to +\infty} \tan{kx}$ a right example?
But I do not know whether this is a function at first. Is this a function?
Could you give a correct real-valued function that is continuous at precisely one point?
Thanks.
 A: The function
$$f(x)=\begin{cases}x\text{ if }x\in\mathbb{Q}\\0\text{ if }x\notin\mathbb{Q}\end{cases}$$
is continuous at $x=0$ but nowhere else.
A: Your example isn't going to work because it actually isn't defined for any nonzero $x$.  If $x$ is nonzero, then $kx$ gets either positively or negatively infinite as $k \to \infty$, and $\tan kx$ is going to zoom around periodically and so will not have a limit.
The standard example of a function continuous at only 1 point is something like the one given by Zev Chonoles, where $f$ takes the value of some continuous function (in this case, $f(x)=x$) on a dense subset of the reals that has a dense complement (in this case, $\mathbb{Q}$), and another continuous function (in this case, $f(x)=0$) on the complement. If the two functions coincide at exactly one point, then you get continuity at that point; everywhere else, the function bounces around crazily between the two continuous functions it is cobbled together from, because it takes each one's value on a dense subset of $\mathbb{R}$.
A: Your example is an expression that defines a function, but the domain of said function is not the real line, unless you allow extended real numbers as the output; but, by the criteria of your problem statement, you do not allow that.If you use the extended real line as a codomain, then the formula that you gave describes a function that maps the entire real line onto the three point set $\{-\infty,0,\infty\}$.  In this case, the function would be continuous at every real number except $0$.
However, at the outset, your problem statement requires that the codomain and range both be the real line.  This is incompatible with the above discussion, because the real line includes neither $\infty$ nor $-\infty$.
