gn.general topology - Continuous function from $[0,1]$ to $[0,1]$

Yes. In fact, there exists such an $f$ taking every value uncountably many times.

Take a continuous surjection $g: [0, 1] to [0, 1]^2$. (Such things exist: they're space-filling curves.) Then the composite $f$ of $g$ with first projection $[0, 1]^2 to [0, 1]$ has the required property.