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  1. Is differentiation $f(x) \rightarrow f'(x)$ a continuous function from $C^1[a,b] \rightarrow C[a,b]$ ?

  2. Is integration $f(x) \rightarrow \int_a^x \! f(t) \, \mathrm{d}t $ a continuous function from $C[a,b] \rightarrow C[a,b]$

where metric for $C$ is the sup metric.

metric for $C^1$ isn't given in the question. Am I able to use the definition of continuous function here then? without knowing metric for $C^1$?

Also, I tried using the fact that for f: M -> N,

f is continuous if and only if preimage of open/closed subset of N is open/closed but am stuck from the very beginning.

thanks.

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  • $\begingroup$ Whether differentiation is continuous depends on the metric (topology) on $C^1$. $\endgroup$ – Daniel Fischer Oct 14 '14 at 19:48
  • $\begingroup$ Think sequences. If $f_n \rightarrow f$ in $C^1[a,b]$, does $f'_n \rightarrow f'$ as well (in $C[a,b]$)? How are the topologies given, if not by a metric? $\endgroup$ – Henno Brandsma Oct 14 '14 at 19:49
  • $\begingroup$ It may help to translate these questions into the language of uniform convergence, since that's what it means to converge in the sup norm. $\endgroup$ – Neal Oct 14 '14 at 19:55
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Assuming both the beginning and target space are taken under the sup-metric:

Note that since both integration and differentiation are linear, it is sufficient to check whether or not they are bounded.

In fact, integration is continuous, but differentiation is not. In particular, take $f_n : [0,1] \to \Bbb R$ to be given by $$ f_n(x) = e^{nx} $$ We note that $\|(f_n)'\|/\|f_n\| = n$.

On the other hand, for any $g:[a,b] \to \Bbb R$, $$ \left\| \int_a^{(\cdot)} g(t)\,dt\right\|_\infty \leq (b-a)\|g\|_\infty $$


Another approach is that which Neal indicates. Consider $f_n = \sqrt{x^2 + 1/n}$. Note that $f_n$ converges uniformly while $f_n'$ does not. On the other hand, $\int_a^x g_n(t)\,dt$ converges uniformly whenever $g_n$ converges uniformly.

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