Hi
My question is about convex functions
I've been reading Boyd's convex optimization and a question came to me. I guess if $f$ is a convex function then $f$ is increasing .
I guess it is conceptually but I didn't find a mathemathical theorem to prove it .
What constraints that we assume makes the gusse be true ?
I would appreciate any counter examples or constraints :)
And I mean by an increasing definition that $f$ is increasing if : $a \ge b$ then $f(a) \ge f(b)$
And also assume that $f$ is continuous
No consider a continuous convex increasing function $f$. Then $g:x\mapsto f(-x)$ is a convex decreasing function. This is because a mirror transformation on points $a$ and $b$ will keep the mirrored graph of $f$ below the mirrored segment $[(a,f(a)),(b,f(b))]$, but obviously a mirrored increasing graph becomes decreasing. For example consider $e^{-x}$.
You can even be non-monotonic. Consider something like $x^2$.
For functions of class $\mathcal C^2(\mathbb R)$, what convexity means though, is that your tangent will all raise. So once it becomes increasing, the function cannot become decreasing later; that would give you a smaller derivative.