Question

Is there a continuous function f:[0,1]~>[0,1] that is not constant in any nontrivial interval such that f^-1{0} is uncountable?

Answer 1

Yes. Just take your preferred non-constant continuous function $g:[0,1]\to[0,1]$

such that $g(0)=g(1)=1$. For instance, you can take $g(x)=x(1-x)$

Now take any Cantor set $K\subset [0,1]$ and, for each interval $(a,b)$ of $[0,1]-K$

for $x\isin (a,b)$ define f to be

$f(x)=g(x-a/b-a)$