# Monotone one-way hash function

I am sitting on a contradictory requirement in which I would need a function that is a one-way hash, but that is also monotone.

One-way: if $y=h(x)$, then computing $x=h^{-1}(y)$ is intractable, or at least, difficult. A textbook example could be SHA-2.

monotone: if $x_1 < x_2$, then $h(x_1) < h(x_2)$. In other words, the hashing would also preserve the order.

With these requirements a bit contradictory, does such function exist already? Would it still be possible to design one, or if not, would it conceivably be possible to construct an impossibility proof that such function cannot possibly exist?

If the order is preserved, then if we are given $h(x)$, we can find $x$ easily by doing binary search.