Continuous Mappings into Hausdorff Space coinciding on Everywhere Dense Set coincide

Theorem
Let $\struct {X, \tau_X}$ be a topological space.

Let $\struct {Y, \tau_Y}$ be a Hausdorff space.

Let $D$ be an everywhere dense subset of $X$.

Let $f : X \to Y$ and $g : X \to Y$ be continuous mappings such that:


 * $\map f x = \map g x$ for all $x \in D$.

Then:


 * $\map f x = \map g x$ for all $x \in X$.