Prove：any smooth map $f:M \rightarrow \mathbb{R}$ can't be one to one. Let $M$ be a connected smooth manifold, $\dim M \ge 2$. Prove：any smooth map $f:M \rightarrow \mathbb{R}$ can't be one-to-one.

No continuous map from $M$ to $\mathbb{R}$ can be injective.
As $M$ has dimension at least $2$ it contains a subspace $C$ homeomorphic to a circle. Now there is no continuous injective map $g$ from $C$ to $\mathbb{R}$. If there were, then there are $a$ and $b$ on $C$ with $g(a)<g(b)$. Then on each arc of $C$ with endpoints $a$ and $b$ there is a point mapped to $\frac12(g(a)+g(b))$ (intermediate value theorem).
