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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 $\Delta$.

Clarinetist
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K.SEN
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  • What do you mean that $f(x)$ is random? From what set of functions is it being drawn? – Théophile Apr 23 '16 at 14:27
  • I've deleted my answer. I misinterpreted the question...Do you mean that $f$ is any function from $[-10,10]$ to $\mathbb{R}$? That's how I interpreted it. But I think you meant fix a function $f$ first. – Jonathan Gafar Apr 23 '16 at 14:29

1 Answers1

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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.

mvw
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