Let $g:[0,1]\to \mathbb{R}$ Suppose $g(0) < g(1)$ and for each x in $\mathbb{R}$ the set $g^{-1}(x)$ consists of at most one element

Question

Let $g:[0,1]\to \mathbb{R}$ Suppose $g(0) < g(1)$ and for each x in $\mathbb{R}$ the set $g^{-1}(x)$ consists of at most one element. Prove that $g$ is strictly increasing.

I would appreciate any help, as I do not know where to start.