Real analysis continuity problem

Question

I have encountered this problem. Can someone help me with it?

Let $f, g : [0, 1] → R$ be two continuous functions, such that $f(x) = g(x), ∀x ∈ [0, 1] ∩ Q$. Prove that $f(x) = g(x), ∀x ∈ [0, 1]$.

Answer 1

Let $x \in [0,1]$. If $x$ rational, then $f(x) = g(x)$ is given. If $x$ is irrational, let $\{q_n\}$ be a sequence of rationals converging to $x$. By continuity, \begin{align*} f(q_n) \to f(x) \\ g(q_n) \to g(x) \end{align*} Since the $q_n$'s are rational, we must have \begin{align*} f(q_n) = g(q_n) \Rightarrow f(x) = \lim f(q_n) = \lim g(q_n) = g(x) \end{align*} for any $x\in [0,1]$.