Continuous functions taking every value a given number of times

We give necessary and sufficient conditions on a function $f:[0,1]\to {0,1,2,…,\omega,\continuum}$underwhichthereexistsacontinuousfunction$F:[0,1]\to [0,1]$suchthatforevery$y\in[0,1]$wehave$|F^{-1}(y)|=f(y)$.