Domain invariance for smooth functions

Question

The domain invariance theorem states that for an open set $U\subset \mathbb{R}^n$ and a continuous and injective mapping $f:U\to \mathbb{R}^n,$ the image $f(U)\subset \mathbb{R}^n$ is open. I've read that for a smooth functions the proof of the analogous statement is easy. Surely, when $f$ is continuously differentiable and for all $x\in U \ $ $\det f'(x)\ne 0$ the image $f(U)$ is open (a consequence of the inverse function theorem). But how to prove that $f(U)$ is open in the case when $f$ is continuously differentiable and injective? Is there an elementary proof?