Baby Rudin continuity problem question

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The discussion revolves around a problem from Baby Rudin regarding the continuity of a real function f defined on R, given that the limit of the difference quotient approaches zero as n approaches zero. It is concluded that this condition does not imply continuity, as it only indicates the absence of simple discontinuities, meaning that the left-hand and right-hand limits at any point are equal. An example is provided where a function can have a simple discontinuity despite meeting the limit condition. The participants clarify the distinction between having equal one-sided limits and actual continuity at a point. The conversation emphasizes the importance of understanding the nuances of continuity in real analysis.
genericusrnme
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Sup guys, I was just going over my Baby Rudin and I came across a problem that I don't really know how to get started on.

Suppose f is a real function defined on R that satisfies, for all x Limit_{n\ \rightarrow \ 0} (f(x+n)-f(x-n)) = 0, does this imply f is continuous?

My first thoughts are that no, it doesn't imply f is continuous, it just implies that f doesn't have any simple discontinuities since f(x_+) = f_(x_-). I don't know how I can go about showing this though..

Could anyone nudge me in the right direction?

Thanks in advance!
 
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hi genericusrnme! :smile:
genericusrnme said:
… it just implies that f doesn't have any simple discontinuities …

does it ? :wink:
 
Well, what, exactly, do you mean by a "simple" discontinuity? If f(x)= 1 for all x except 0 and f(0)= 0, that looks like a pretty simple discontinuity to me!
 
tiny-tim said:
hi genericusrnme! :smile:


does it ? :wink:

Ah yes, you're completely right
f(x+) = f(x-) but f(x+) isn't necessarily equal to f(x)

HallsofIvy said:
Well, what, exactly, do you mean by a "simple" discontinuity? If f(x)= 1 for all x except 0 and f(0)= 0, that looks like a pretty simple discontinuity to me!

Yep, I just got that

Thanks guys!
 

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