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Let $f_n$ and $f$ be continuous functions on an interval $[a, b]$ and assume that $f_n\to f$ uniformly on $[a, b]$. Pick out the true statements:

(a) If $f_n$ are all Riemann integrable, then $f$ is Riemann integrable.
(b) If $f_n$ are all continuously differentiable, then $f$ is continuously differentiable.
(c) If $x_n\to x$ in $[a, b]$, then $f_n(x_n)\to f(x)$.

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c is true.but what about a and b –  poton Aug 9 '12 at 11:54
Again the mere copying of an exercise assigned to you, again no indication whatsoever of what you know, what you tried, and so on. –  Did Aug 9 '12 at 12:35
Since you are new to this site, you could maybe read this: How to ask a homework question. I wrote this comment because the question sounds homework-like. Let me also add that I don't think that multiple choice-questions are best form of questions for Q&A site. –  Martin Sleziak Aug 9 '12 at 13:59

2 Answers 2

HINT for (b): Can you find a sequence of continuously differentiable functions on $[-1,1]$ converging uniformly to $|x|$?

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please explain. –  poton Aug 9 '12 at 12:14
@poton: To think about finding such a sequence, draw some pictures. The point is that although $f(x)=|x|$ is continuous, it’s not differentiable at $0$. –  Brian M. Scott Aug 9 '12 at 12:18
but in b fn are continuously differentiable.does there exist such functions which converges to |x| –  poton Aug 9 '12 at 12:21
and what about a –  poton Aug 9 '12 at 12:22

(a) is true. Continuity on closed interval implies Riemann integrability (see its equivalent characterization). Note that the assumption about continuity of limit function is unnecessary, as uniform convergence provides it.

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