inverse of a non decreasing step function, defined as an inf $g : \mathbb R \to [0,1]$ is a non-decreasing and right continuous step function such that $g(x)=0$ for all $x \leq 0$ and $g(x)=1$ for all $x \geq 1$. Let us define $g^{-1}(y) = \inf { \{x : x \geq 0, \ g(x) \geq y\} }$
Then, is it a continuous function, right/left continuous or neither ?
Where I'm specifically having a problem is the [$g(x) \geq y$]  part. I do not understand what this means in this context.
Edit: it has been pointed out to me that the function $g$ is not defined in $(0,1)$ so the question is incorrect. So please just assume that function is well defined but however many steps the question says exists, exist between $(0,1)$.
 A: Try it on the example $ g(x) = \begin{cases}
          0, & x < 0,\\
          1, &  x \geq 0
          \end{cases} $ and follow the definition to see what is happening in this definition.  You should be able to explicitly calculate $ g^{-1} $.  In particular, calculate the five cases $ g^{-1}(x) $ for $ x < 0 $, $ x = 0 $, $ 0 < x < 1 $, $ x = 1 $, and $ x > 1 $.
A: I understand. I think whoever edited the question rewrote the definition of the inverse function and Using it on an easier example given by jake helped me understand the definition.
Eg  When $y=-1$, then $g^{-1}(-1)=\inf\{x : x\ge0, g(x) \ge -1\}$ so we just take $g(x)$ as $0$ since no value of $x$ maps to $y= -1$ under $g$.
Now the solution. Can someone please confirm this
lets assume there are steps in this function for simplicity
    $g(x) = 0$ if $x <= 0$

    $g(x) = a$ if $0 < x < (1-e)$  where 0<a,e<1

    $g(x) = 1$ if $(1-e) < x$

this "e" is important since it is given that $g$ is right continuous and $g(x) =1$ for $x\ge 1$ This means the beginning of the step should lie to the left of 1
we then calculate inverse and if $y$ belongs to (-$\infty$, a] then $g^{-1}(y)= 0$
for $y$ belongs to $( a, 1], g^{-1}(y)= (1-e)$
For $y$ greater than $1$, no $g(x)$ exists that can satisfy the inequality so inf is a null set.
Clearly, the inverse function is left continuous. Ideally, like jake said, this should be split in 5 cases but since it came out to be same ive shortened it
