It's useful to think of these spaces in terms of natural topologies on
them, and continuity etc.

note that differentiation is an unbounded operator!
just look at $x^n$ on $[0,1]$
this makes it rather trickier to analyze...

push the uniform norm forward by $f'$ (i.e., onto $C^1$ modulo
constant functions) and you get $\norm{f}=\sup f - \inf f$.
  (for a function on $[a,b]$)
With respect to this norm,
$\norm{f'}_\infty \geq \frac{1}{b-a}\norm{f}$
by MVT (mean value theorem)
...which in particular shows that
 $I_a\from C^0 \to C^1$ is a bounded (hence continuous) operator,
 in contrast to differentiation.
