According to the intermediate value theorem, if $f$ is a real-valued continuous function on the interval $[a, b]$, and $u$ is a number between $f(a)$ and $f(b)$, then there is a $c ∈ [a, b]$ such that $f(c) = u$.
Now consider the function:
$$f(x) = x^3 - \lceil x \rceil$$
This function is obviously not continuous - it jumps whenever $x$ is integer. However, the jumps are always downward. Therefore, if $a<b$ and $f(a)<u<f(b)$, then there is a $c ∈ [a, b]$ such that $f(c) = u$ . I don't know how to prove it formally, but intuitively, the function cannot jump upwards, so in order to go from $f(a)$ up to $f(b)$, it must pass through all points in between.
My question is: Is there a theorem like this? If so, what is its name and what reference can I use for it?