Let $g : \Bbb R \to [0, 1]$ be a non-decreasing and right-continuous step function such that
$$ g (x) := \begin{cases} 0 & \text{if } x \le 0\\ 1 & \text{if } x \ge 1\end{cases}$$
Let us define $g^{−1}$ as follows,
$$ g^{-1}(y) := \inf \left\{ x \ge 0 \mid g(x) \ge y \right\} $$
The answer provided to me is that $g^{-1}$ is left-continuous but not right-continuous.
I think that the question is wrong because the function is not defined for values in the interval of $(0,1)$. Can anyone please help me with this?