Can a Sequence of Step Functions Uniformly Converge to a Continuous Function?

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Discussion Overview

The discussion centers around the question of whether a sequence of step functions can uniformly converge to any continuous function. The scope includes theoretical aspects of analysis, particularly in relation to Riemann integrability and uniform continuity.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes the importance of uniform convergence for defining a non-ambiguous integral of a function, contrasting it with pointwise limits.
  • Another participant mentions that not all Riemann integrable functions are regulated, highlighting that a regulated function has a countable set of discontinuities.
  • A specific example is provided of a function defined on the Cantor set, which is Riemann integrable but not regulated, raising questions about the properties of continuous functions.
  • It is proposed that continuous functions on a compact interval are regulated, with uniform continuity being a key aspect of this argument.
  • One participant questions the necessity of uniform continuity for the function in question, suggesting that the compactness of the interval might suffice.
  • A later reply confirms that continuity on a compact set implies uniform continuity, addressing the earlier question.

Areas of Agreement / Disagreement

Participants generally agree on the relationship between continuity, compactness, and uniform continuity, but there is some debate about the necessity of uniform continuity for the convergence of step functions to continuous functions.

Contextual Notes

The discussion does not resolve whether uniform continuity is strictly necessary for the convergence of step functions to continuous functions, leaving this as an open question.

"pi"mp
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This isn't a homework question. My adviser has me studying basic analysis and has lately pushed me towards the following question:

"Let f be any continuous function. Can we prove that there exists a SEQUENCE of step functions that converges UNIFORMLY to f?"

I have noticed this idea is important to Riemann integrability but I can't seem to prove this. Or notice why the idea of uniform convergence is so key.

Any help? Thank you
 
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Functions which are the uniform limit of step functions are usually called "regulated functions". The uniform limit is important as it allows us to define a non-ambiguous integral of such a function. We might not do the same for pointswize limits.

Not all Riemann integrable functions are regulated. Indeed, it is a theorem that the set of discontinuities of a regulated function is countable. It is also known that a function is Riemann integrable iff it's set of discontinuities has Lebesgue measure zero.

Now, the Cantor set C has Lebesgue measure zero, but is not countable. So the function

f:\mathbb{R}\rightarrow \mathbb{R}:x\rightarrow \left\{\begin{array}{c} 1~\text{if}~x\in C\\ 0~\text{if}~x\notin C \end{array}\right.

is Riemann integrable but not regulated.

Your question now asks whether a continuous function f:[a,b]\rightarrow \mathbb{R} is regulated. This is indeed the case.
A key point here is that f is uniform continuous. So we can determine for each epsilon a delta such that

|x-y|<2\delta~\Rightarrow~|f(x)-f(y)|<\epsilon

Now, how do we approximate f by step functions. Well, let [a,b]=[a,a+\delta]\cup [a+\delta,a+2\delta]\cup ...\cup [a+n\delta,b] be a "partition" such that each part has length <\delta. Then we define

g(x)=g(a+k\delta)~\text{if}~x\in [a+k\delta,a+(k+1)\delta]

Then the uniform distance between g and f is smaller than epsilon.

I'll leave the details and verification to you!

Do not that basically all functions we need in our daily life will be regulated.
 
Last edited:
Micromass beat me to it :( But to add one small point to his post, it is important to note that the compactness of [a,b] is crucial.
 
micromass, why is it important that f itself be uniformly continuous? That wasn't one of the assumptions I was given. Shouldn't this work even if f itself is non-uniformly continuous? Thanks for your reply.
 
ah I think I answered my own question. It is because it is defined on a compact set, correct?
 
"pi"mp said:
ah I think I answered my own question. It is because it is defined on a compact set, correct?

Yes: continuous on a compact set is uniform continuous.
 

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