Suppose $A$ is a countable subset of $\mathbb{R}$. Prove that there exists a continuous function $\phi$ from $A$ to $A^c$

Since $A$ is countable, $ M := \{ x-y~~|~~x, y \in A \}$ is countable. Hence there exists a $z \in M^c$.

Let $\phi: A \rightarrow A^c$, $\phi(x) = x+z$.

$\phi$ is injective and continuous.