$f|_{(a, b)}$ convex. Is $f$ convex on $[a, b]$?

Question

Suppose $f : [a, b] \rightarrow R$ is continuous on $[a, b]$ and convex on the open interval $(a, b).$ Show that $f$ is convex on the closed interval $[a, b].$

Answer 1

It is enough to check that $f\bigl(tx+(1-t)y\bigr)\le tf(x)+(1-t)f(y)$ for any $t\in[0,1]$ and $x\in[a,b]$. If $x,y\in(a,b)$, then we have it by convexity. If, for instance, $x=a,y\in(a,b)$, then take $x_n=a+\frac{1}{n}$ and pass to a limit with $n\to\infty$. The remaining cases are handled in a similar way.

The statement remains true, if $f(a^+)\le f(a)$ and $f(b^-)\le f(b)$. These limits do exist, because $f$ is either monotone, or unimodal in the sense that $f$ decreases (possibly weakly) on $[a,c]$ and increases (possibly weakly) on $[c,b]$ for some $c\in(a,b)$. The monotonic functions admit one-sided limits.