There exists $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $\mathrm{grad} f = g$.

Question

I want to know whether this is true :

For all function $g : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ of class $\mathcal{C}^1$ there exists $f : \mathbb{R}^2 \rightarrow > \mathbb{R}$ such that $\mathrm{grad} f = g$.

I can't prove it is true, but I can't prove it is false with a counter-example either. Thanks for the help.