Recall a topological space $X$ is *path connected* if for all $x,y \in X$ there is a continuous function $f\colon [0,1] \to X$ such that $f(0)=x$ and $f(1)=y$.

Say that $X$ is *continuously path connected* if there is a continuous function $f\colon [0,1] \times X \times X \to X$ such that $f(0,x,y) = x$ and $f(1,x,y) = y$

Is every Polish path connected space also continuously path connected.

(I'd like to know the answer for Polish spaces, but if it is positive, feel free to mention the answer for larger classes of spaces.)

My guess is that the answer is no, and a counterexample will involve constructing a space with uncountably many "forks in the road" where deciding whether one can go left or right is not continuous in the initial data. However, I assume something like this has a well-known counterexample, so I am asking it here.

(Feel free to change the tags. Is this homotopy theory?)