(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have to show that f: R->R is a step function if and only if:

1) f is continuous except at finitely many points of R

2) f takes only finitely many distinct values

3) f(x) -> 0 as |x| -> infinity

2. Relevant equations

3. The attempt at a solution

I think I have shown that assuming f is a step function then 1, 2 and 3 hold.

However, I'm not sure about going the other way around, if 1,2 and 3 hold then f is a step function. The question advises using another theorem that I should have learnt last term, but it doesn't specify which and I can't figure out which it means. I was studying differentiation in analysis last term.

Using 3, I can show there exists an a0 such that f=0 for x<a0 and an such that f=0 for x>an

EDIT: I'm not so sure that this is as simple as I initially thought. With just 3 on it's own it may never reach 0.

Does 2 imply that f must be constant over a finite number of intervals? This doesn't seem very rigorous. I suppose this also uses 1 that f is continuous?

Thank you :)

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# Homework Help: Priestley step functions

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