how do I find supremum of the set

Question

I have a test on Monday and the professor gave a hint about the problem. f : ??? is random How do I solve this problem?

Let $f: [-10, 10] \to \mathbb{R}$ be defined by $f(x) = ???$. Let $\Delta$ be the set of positive numbers such that $$\Delta = \{\delta: |x-y| < \delta \implies |f(x)-f(y)| < 1\}\text{.}$$ Find the supremum of the set $\Delta$.

Answer 1

IMHO there is no specific answer to the general problem as it would need properties of the given $f$.

Try a couple of different functions, e.g. $f(x) = 42$, $f(x) = x + 3$, $f(x) = 3x^3-2$, $f(x) = (x-4)/(x+2)$, $f(x) = \sin(x)$, $f(x) = \int_3^x \lfloor \log(y) \rfloor^2 / \sqrt{y}\, dy$ or whatever to get an idea.