Let $f$ be a weakly increasing step function. $g(x)=ax$ be a weakly increasing linear function.
Let $h=f+g$. What is $h^{-1}$?
$f+g$ is not continuous. However, at a first glance, it seems possible to intuitively "define" $h^{-1}$ to be continuous! One example function $h^{-1}$ looks like this:
$h^{-1}(x)=x$ when $x\in[0,1]$
$h^{-1}(x)=1$ when $x\in[1,2]$
$h^{-1}(x)=x-1$ when $x\in[2,3]$ etc.
Do we have a name for this class of continous function $h^{-1}$?
On
An inverse function cannot be constant on some interval. This is easy to see graphically in the single variable case as the inverse function is merely the reflection about the line $y=x$. If the graph of an inverse function is constant on a continuous region then when we reflect about the line this section becomes vertical so it isn't a function as it fails the vertical line test. In this case you can see that $h(1)$ isn't well-defined if $h(x)^{-1}=1$ for $x\in [1,2]$.
It looks like what you've called $h^{-1}$ is a function such that $h^{-1} \circ h = 1$. Sometimes, such functions are called retractions.
As the first comment on the OP pointed out, we shouldn't really use $h^{-1}$ to represent the function you defined because, although $h^{-1} \circ h = 1$, $h \circ h^{-1}$ isn't even defined. By definition, an inverse of a function $f : A \to B$ is a function $f^{-1} : B \to A$ such that $f \circ f^{-1} = 1_B$ and $f^{-1} \circ f = 1_A$.