TITLE: Existence of Continuous Function respecting points
QUESTION [1 upvotes]: Suppose that $(X,d)$ and $(Y,d)$ are metric spaces between which there exists a continuous function.  Fix $x \in X$ and $y\in Y$.  When does there exist a continuous function sending $x$ to $y$?

REPLY [1 votes]: Define $f(z):=y$ for all $z \in X.$ Then $f$ is continuous anf $f(x)=y.$
{"set_name": "stack_exchange", "score": 1, "question_id": 3215917}