Under what conditions will $f(x)^n$ converge pointwise and uniformly?

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Let $f(x)$ be a continuous function on $[0,1]$. Under what conditions on $f$ will the sequence $f_n(x)$ = $(f(x))^n$ converge uniformly and pointwise?

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To get you started: What if $|f(x)|>1$ for some $x?$ What if $|f(x)|<1$ for all $x?$