Let $X$ be a compact Hausdorff space. A continuous  map $f:X \to X$ defines a bounded  linear operator  $T_{f}$ on Banach space $C(X)=\{\phi:X\to \mathbb{C} \mid \phi \; is \;\;\; \text{continuous}\}$  with $T_{f}(\phi)=\phi \circ f$.

Put $X=[0,\;1]$

>What is  an example of  a non constant map $f$ such that $T_{f}$ is  a compact operator?
>What is  an example of  a map $f$ such that $T_{f}$ is  a Fredholm operator of non zero index?


For  a  related question [see this  MSE  post](http://math.stackexchange.com/questions/1373804/non-injective-continuous-maps/1373823#1373823)