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