Suppose that $x \in R^m, y \in R^n, z(x) \in R^n$, and $f(x,y)$ is convex in $(x,y)$.
Is $f(x,z(x))$ a convex function in $x$ for arbitrary continuous functions $z(x)$?
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Here is a counterexample for $m=n=1$. Let $f(x,y)=x^2+y^2$ which is convex, and let $z(x)$ be continuous such that $z(0)=0$, $z(1)=2$, $z(2)=1$. Then $$ 2f(1,z(1))=10>0+5=f(0,z(0))+f(2,z(2)), $$ hence $f(x,z(x))$ is not convex.