Baby Rudin continuity problem question

[genericusrnme]

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!

 

[tiny-tim]

hi genericusrnme!

… it just implies that f doesn't have any simple discontinuities …

does it ?

 

[HallsofIvy]

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!

 

[genericusrnme]

hi genericusrnme!


does it ?

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

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!