Consider $M$ a smooth manifold, and $g:M\to\mathbb{R}$. I want to prove the following equivalence:
$dg_x:T_xM\to\mathbb{R}$ is surjective, if and only if $dg_x\neq 0$.
That surjective implies $dg_x\neq 0$ is straightforward. I care about the other direction. My attempt at a proof would go as follows:
Let $r\in \mathbb{R}$ be given. I need to construct, explicitly, in terms of $r$ an $X\in T_xM$, such that $r=dg_x(X)$. My assumption is that $dg_x\neq 0$, so I get a strong feeling that the implicit function theorem should come into play: namely, I can assume that $\dfrac{\partial g}{\partial x^\mu}\neq 0$. I am lacking the details of proceeding.