If a function is convex on some interval $[a, c)$ and on some interval $(c, b]$, under what circumstances is it convex on $[a,b]$?

Question

Suppose we have $a, b, c ∈$ $\mathbb R$, $a < c < b$, and a continuous function $f:[a, b] \rightarrow \mathbb R$.

We know $f(x)$ is twice differentiable for all $x ∈ [a,c)$ ∪ $(c, b]$ and $f(x)$ is not twice differentiable for $x = c$. We also know $f$ is convex on $[a, c)$ and convex on $(c, b]$.

Under what circumstances is $f$ convex on $[a,b]$? Is there any way to check its convexity on $[a, b]$ without the use of Jensen's inequality?

Answer 1

Hint

As $f$ is differentiable in $[a,b]\backslash \{c\}$ can consider the two cases:

• $f'(c^-) \geq f'(c^+)$
• $f'(c^-) < f'(c^+)$