I want to prove that there does not exist any surjective $C^1$ function from $\mathbb{R}^3$ to $\mathbb{R}^5. $
Attempt: I was trying to use the Sard's theorem. If at all such a function exists then the measure of critical values should be zero. After that how to get a contradiction, I am not sure.
The Sard theorem says that if $X$ is the set of critical points of $f$ then $f(X)$ has measure $0$ in $\Bbb R^5$. If $x\in \Bbb R^3$, the differential of $f$ at $x$ is a linear map $$d_xf:\Bbb R^3\longrightarrow \Bbb R^5$$ so $d_xf$ can't be surjective which means that every point of $\Bbb R^3$ is a critical point of $f$, i.e $X=\Bbb R^3$. Hence $f(\Bbb R^3)\not=\Bbb R^5$ because $f(\Bbb R^3)$ has measure $0$ in $\Bbb R^5$.