Typical continuous functions are not iterates (Q921130)

The author proves that the set \(\{f\in C^ 1_ 0:\) \(f=\phi^ k\), \(\phi \in C^ 1_ 0\), \(k\geq 2\}\) is of the first category in \(C^ 1_ 0\), where \(C^ 1_ 0\) denotes the set of continuous functions mapping [0,1] into itself.