# continuous function from one metric space to another metric space

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.

• Whether differentiation is continuous depends on the metric (topology) on $C^1$. – Daniel Fischer Oct 14 '14 at 19:48
• 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? – Henno Brandsma Oct 14 '14 at 19:49
• 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. – Neal Oct 14 '14 at 19:55

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.