Let $f:X \to Y $ a continuous function, where $(X, \tau_X)$ is $T_2$ and first countable and let $D$ dense in X, then $f(D) \cap f(X-D)= \emptyset$
Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism