(1) $X= (0,1]$ and $Y = [0,1]$;
(2) $X = [0,1]$ and $Y$ is the topologist's sine curve
(3) $X = [0,1] \cup [2,3]$ and $Y = X \times X$
I believe there is a continuous function for the first one.
I know there doesn't exist a continuous function for the second one, because $X$ is path connected and the topologist's sine curve is not. As well as, $f(0)$ doesn't exist in $Y$.
I feel like there is no continuous function for (3) because of the disconnection; however, I could be wrong.