a) Give an example where $$ \phi (x) = \sup_{n \in Z^+} f_n(x)$$ is not continuous.
b) Prove that if one assumes that the sequence {$f_n$} is an equicontinuous family, then $\phi : [0,1] \rightarrow R$ defined in part a) is continuous.
My professor provided an example where $f_n(x) = -(\sin(\pi x))^n$ However, I do not see why this example is not continuous and I'm lost for part b).