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a problem from a book

I've been trying to use the definitions of continuity in terms of open sets but nothing has worked so far. Any pointers would be much appreciated. Thanks!

Xander Henderson
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NaturalLog420
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    Do you know anything about Banach spaces? All three of these can be argued directly, but these problems lend themselves very nicely to the language of bounded linear operators. – Xander Henderson Oct 18 '17 at 03:34
  • We haven't covered Banach spaces yet, how could they be argued directly? – NaturalLog420 Oct 18 '17 at 03:35
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    For part (i), you want to show that $C^1$ contains all of its limit points. Start with a sequence of functions in $C^1$ that converges in $C$, then show that the limit function is in $C^1$. – Xander Henderson Oct 18 '17 at 03:51
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    Alternatively, find a counterexample. That is, find a sequence of differentiable functions in $C$ that converges to something that is not differentiable. – Xander Henderson Oct 18 '17 at 04:03
  • With respect to (ii) and (iii), my recollection is that integration is continuous (basically, integration "smooths things out", which gives you a nice way of bounding things), but that differentiation is not continuous with respect to the sup-norm. – Xander Henderson Oct 18 '17 at 04:08

1 Answers1

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i) No. Counter example: Let $f_n(x)=\sqrt{x^2 +\dfrac{1}{n}}$ be a sequence of function in $C^1[-1,1]$ which converges to $f(x)=|x|$ (does not belongs to $C^1[-1,1]$).

ii) No. Counter example: Let $f_n(x)=\frac{x^{n+1}}{n+1}$ be a sequence of function in $C^1[0,1]$ which converges to $f(x)=0$. But $\frac{df_n(x)}{dx}$ does not converge to $\frac{df(x)}{dx}$.

iii) Yes. Let $f_n(x)$ be any sequence converging to $f(x)$. Now $\| \int_a^x f_n(x)dx - \int_a^xf(x)dx\|_\infty$$\leq(b-a)\|f_n(x) -f(x)\|_\infty$ tends to zero. Hence the function is continious.

Nirmal Kotal
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