Let $f:C\to [0,1]$ and $C$ is the Cantor set with subspace topology $\tau_C$ and $f$ is the Cantor function, i.e. take the ternary only with 0,2 on bits and maps to binary with 0, 1.
How can I show $K\subseteq[0,1]$ is Borel if and only if $f^{-1}(K)$ is? Actually, I want to show with the subset topology, i.e. for all $K\in \mathfrak{B}(\tau_{[0,1]})$, if and only if $f^{-1}(K)\in \mathfrak{B}(\tau_{C})$.
And is this equivalent to $f(K)\subseteq[0,1]$ is Borel if and only if $f(K)$ is?
That's because the Cantor function is a measurable function. Every continuous function is measurable, every monotone function, too: for a continuous function, the pre-image of an open set is open, for a monotone function, the pre-image of an interval is an interval. Both intervals and open sets generate the $\sigma$-algebra of Borel sets. And the Cantor function is both monotone and continuous.