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I wonder the following property hold but I am unable to prove it. Can anyone help me !

Given functions $f,g: \mathbb{R}\to \mathbb{R}$. Assume that $f$ be piece-wise linear and g is convex/concave. What kind of conditions on function $f$ to make composition function $g\circ f$ is convex/concave?

Its easy to check if $f$ is linear with second derivative but here the problem is that function $f$ is not differentiable.

Thomas Edison
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  • What have you tried? What have you in mind? Are there finitely many pieces? A simple necessary conditions is, that $g\circ f$ is continuous. – user251257 Oct 22 '16 at 16:31
  • I'm not a fan of the question the way it's worded. After all, any conditions that you might suggest for $f$ are sufficient but not necessary. That said: have you read Boyd & Vandenberghe's Convex Optimization, particularly the section on convex functions? You'll have a very clear answer there. – Michael Grant Oct 23 '16 at 16:36

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