I have been studying the continuity of a convex function and having a trouble below:
In some books, the authors defined the the continuity of a convex function $f$ even $f$ is not in $\mbox{dom }f$, for example $$f(x)=\frac{1}{x} \mbox{ if } x>0, f(x) = +\infty \mbox{ if } x\leq 0,$$ is continuous on $\mathbb R$. This way may cause some unusual thought such as if $f$ is continuous at $x$ then $f$ is bounded on a neighborhood of $x$ will no longer be true.
My question is what do we need to define the continuity at points where the value of function is $\pm \infty$ for?
If $f(a)=+\infty$, we can say that $f$ is continuous at $a$ in the extended sense if for every $M\in\mathbb R$ there is $\delta>0$ such that $f(x)>M$ whenever $|x-a|<\delta$.
Notice this is not something special to convex functions. It is often convenient to extend the notion of continuity in this way.
A higher level explanation is that the extended real line $\overline{\mathbb R} = \mathbb R \cup\{-\infty,+\infty\}$ can be given a topology (and even a metric); then we just use the topological definition of continuity for functions taking values in $\overline{\mathbb R} $.