Let $M: C([0,1]) \rightarrow C([0,1])$ be defined by $$ Mf(x) = f(x/2), \;\; x\in[0,1]$$
Prove that the range of $I-M$ does not contain nonzero constant functions, but it contains all functions $C([0,1])$ that are differentiable (from the right) at 0 and satisfy $f(0) = 0$
My try: if $(I - M)f = const$ then the derivative $f'(0)$ would be infinite and hence the function not continuous. By this it also seems to follow that differentiability at 0 and f(0) = 0 must be necessary conditions for $g \in R(I-M)$ (R is the range). But how can I prove that we can reach all such functions?