Let $f : \mathbb{R}^n→ \mathbb{R}_∞$ be convex over the sets A, B which are also convex.
$A ∩ B = ∅$ and $A ∪ B$ is convex.
Then is $f$ is convex on $A ∪ B$? Why or why not?
I am confused particularly by the meaning of a function being convex on a set.
My intuition tells me that $f$ is indeed convex on $A \cup B$ but I can't put my thoughts down on paper in writing.
$A=[0,1]$ and $B=(1,2]$ are convex. $f(x)=0$ for $x \neq 1$ and $ f(1)=1$. What do you think of this case?