Suppose I have a differentiable function $f:[a, b]\rightarrow R$, with $a,b\in I\!R\cup\{-\infty,\infty\}$, with $\lim_{x\rightarrow a}=0$ and $\lim_{x\rightarrow b}=1$. Is there any algorithm to determine if $f$ is nondecreasing?
What about particular cases of the problem (Rationals, Integers, limited or finite image - finite domain is trivial, any other nontrivial set)?