Functions continuous at only one point - more exotic examples? The canonical example of a function continuous at only one point is 
$$f(x) = \chi_{\mathbb{Q}}(x) \cdot x$$ which is continuous only at $0$. 
A user on another question pertaining to this issue has also stated that if $g(x)$ is a bounded and nowhere continuous function, then $f(x) = x \cdot g(x)$ works. 
Do we have more exotic, nontrivial examples dissimilar to ones of this form? What are some necessary conditions? 

A function that is only continuous at $x_0$ with $f(x_0) = y_0$ is always of the form $f(x) = (x-x_0)g(x) + y_0$, where $g$ is nowhere continuous. Just define $g(x) = \frac{f(x)-y_0}{x-x_0}$ for $x\neq x_0$ and $g(x_0) = $whatever you want so that $g$ is not continuous at $x_0$.
