A question about partial inverses of a map from the Cantor set

Question

Let $X = \{0,1\}^\mathbb{N}$ with the metric $d(x,y) = 2^{-\text{min}\{i|x_i\neq y_i\}}$. It is known that $X$ is homeomorphic to the standard Cantor set. Define a map $f:X\to \mathbb{R}/\mathbb{Z}$ $$f(a_1a_2\dots) = \sum_{i=1}^\infty a_i2^{-i}$$ How can I show that if $g:\mathbb{R}/\mathbb{Z}\to X$ is any map such that $f(g(x)) = x$ for all $x \in \mathbb{R}/\mathbb{Z}$ then $g$ is not continuous? I can see that $f$ is continuous but not invertible, and that there cannot be a homeomorphism between the two spaces, but $g$ is not necessarily an inverse. Any help would be greatly appreciated.

Answer 1

Following Alessandro's hint: since connectedness is a topological property a continuous function $g$ would have to send the entire connected domain to a singleton i.e. be constant, which contradicts the stated property..