Let $g : X \rightarrow Y$ be a homeomorphism, and let $f : Y \rightarrow \mathbb{R}$ be a continuous function. Then the extremes of $f$ in $Y$ are mapped to the extremes of the composition $f \circ g $ in $X$, and if $f$ has a unique minimizer in $Y$, then $f \circ g $ has a unique minimizer in $X$. This seems plausible to me, but does anyone know of a reference where this is proved (and preferably related results)?
Kind regards
Olav