No continuous mapping

Prove that there does not exist an $1-1$ and continuous mapping from $\mathbb{R}^2$ to $\mathbb{R}$ .

Solution

Due to connectness we have that $f \left(\mathbb{R}^2 \right)= \mathcal{I}$ where $\mathcal{I}$ is an interval. Note that if we remove a point from the plane, it still remains connected. Having that in mind we observe that $f \left ( \mathbb{R}^2 \setminus f^{-1}\left ( a \right ) \right )$ is connected and equals $\mathcal{I} \setminus \left \{ a \right \}$ whereas this is not connected leading to a contradiction.