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Pick out true statements

1.Any continuous function on $[0,1]$ is of bounded variation

2.If $f$ is continuously differentiable on $\mathbb R$the restriction of $f$ on $[-1,1]$ is of bounded variation

3.Any monotone function is of bounded variation.

I am able to show 3 is true; for 1 I have the following counter example but not sure about the other one

$f(x)=xsin\frac{1}{x} ;x\neq0 $

$f(0)=0$

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  • $\begingroup$ On a purely logical argument, if all continuous functions were BV, then we would not need a name for BV functions. Think of $ f(x)=sin(1/x), x\neq 0, f(0)=0$. For 2, yes, continuous on compact is uniformly continuous. $\endgroup$ – Passing By Dec 6 '14 at 6:55
  • $\begingroup$ You do not need that "continuous on compact is uniformly continuous". Still, a continuously differentiable function on a compact interval is even absolutely continuous an in particular of bounded variation. $\endgroup$ – PhoemueX Dec 6 '14 at 8:03
  • $\begingroup$ @PassingBy: Even if all continuous functions are of BV, there are still noncontinuous functions of BV. Also, your example $f$ is not continuous on $[0,1]$. $\endgroup$ – user99914 Dec 6 '14 at 8:13
  • $\begingroup$ @PhoemueX: I never claimed it was necessary, only that it is sufficient. $\endgroup$ – Passing By Dec 6 '14 at 22:58
  • $\begingroup$ @John: Sorry, I just meant that uniformly continuous functions are BV, and I misread your claim, sorry. And, yes, I missed the term x in xsin(1/x) , my bad. $\endgroup$ – Passing By Dec 6 '14 at 23:05

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