There are examples of functions $f \colon [0,1] \longrightarrow [0,1]$ such that for any $\alpha $, $f^{-1}(\lbrace \alpha \rbrace)$ is uncountable. My favorite example is $$f(r) = \limsup_n \frac{a_1 + a_2 +\cdots + a_n}{n}$$ where $0.a_1a_2\cdots$ is the (non-terminated) binary expansion of $r$.

Is there a continuous function $f \colon [0,1] \longrightarrow \Bbb{R}$ such that for any $\alpha \in \Bbb{R}$, $f^{-1}(\lbrace \alpha \rbrace)$ is (non-empty) perfect ?