If $f$ is convex function and $f$ is continuous on $(a,b)$.When is $f$ continuous at $a$ and $b$?

Question

Let $f:[a,b]\to \mathbb{R}$. We know that if $f$ is convex function then $f$ is continuous in $(a,b)$. Ιn which cases is $f$ continuous at $a$ and $b$?

Answer 1

I think the best you can do, without knowing more about $f$, is say that $f:[a,b]\rightarrow\mathbb{R}$ is continuous at $a$ if $f(a)=\lim_{x\to a^+}f(x)$ and likewise for $f(b)$. For example, consider $f:[0,1]\rightarrow\mathbb{R}$ s.t. $f(0)=1$, $f(x)=-\sqrt{x}$ for $x>0$. Then f is convex on $[0,1]$ but $f$ is not continuous at its left endpoint.

Answer 2

First, if $f$ is convex on $(a,b)$ then finite or infinite limits $\lim_{x \to a+} f(x)$ and $\lim_{x \to b-} f(x)$ always exist.

Consider the functions $f:x \mapsto x^2$ on $[0,1]$ and

$$g(x) = \begin{cases}f(x), & 0 \leqslant x < 1,\\ 2, & x= 1 \end{cases}$$

Both are convex, but $f$ is continuous at $x=1$ and $g$ is not.

What does this tell you?

The one-sided limit at an endpoint is always a finite number or $\pm \infty$. The limit cannot fail to exist (as with an oscillating function). As long as limits are not infinite, the function may or may not be continuous at the endpoints, but it can be made continuous by changing values to match the one-sided limits.