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.

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.