# Optimizing a Composed Function

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ be continuous.

Consider $g: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ strictly increasing, continuous and such that $g(0)=0$.

I think this is an interesting (maybe trivial) question:

are $f(\cdot)$ and $g(f(\cdot))$ sharing the same minima?

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For all $x$ and $y$ in $\mathbb R^n$, $f(x) \le f(y)$ if and only if $g(f(x)) \le g(f(y))$, so yes. –  Rahul May 10 '12 at 0:47