Fixing a complete Riemannian metric g on $\mathbb R^n$, we show that a local diffeomorphism $f : \mathbb R^n\to \mathbb R^n$ is bijective if the height function $f\cdot v$ (standard inner
product) satisfies the Palais-Smale condition relative to $g$ for each for each nonzero
$v\in \mathbb R^n$. Our method substantially improves a global inverse function theorem of
Hadamard. In the context of polynomial maps, we obtain new criteria for invertibility
in terms of Lojasiewicz exponents and tameness of polynomials.