We have that $X, Y$ are metric spaces and $f, g: X \rightarrow Y$ are continuous functions such that $f(x) = g(x)$ $\forall\ x \in E$, where $\bar{E} = X$, and I have to prove that $f = g$.
I understand that, for $f$ and $g$ to be equal, the equality in E needs to imply that the equality holds in $E'$ as well using the continuity hipothesis, but I am a little bit unsure...
My first idea was to use the squential definition of continuity, maybe showing that: if $(x_n)_n$ is a sequence in $E$ that converges to a point $x_o$ in $E'$, that exists by the definition of $E'$, thus $(f(x_n))_n$ and $(g(x_n))_n$ are so that $f(x_n) = g(x_n)$ for all $n \in \mathbb{N}$, and by the continuity hipothesis $\lim f(x_n) = f(x_o) = \lim g(x_n) = g(x_o)$.
This is a sketch of my very first decent attempt, is it going in the right direction? Am I getting something wrong?
