Two paths $f$ and $f'$ mapping the interval $I=[0,1]$ into $X$, are said to be path homotopic if they have the same initial point $x_0$ and the same final point $x_1$, and if there is a continuous map $F:I\times I\to X$ such that $$F(s,0)=f(s)$$ $$F(s,1)=f'(s)$$ $$F(0,t)=x_0$$ $$F(1,t)=x_1$$ for each $s\in I$ and each $t\in I$.
My question is why is it required that $F$ be continuous on $I\times I$ and not separately continuous? The idea is that $F$ being only separately continuously means that for each fixed $t$ we have that $g(s)=F(s,t)$ is continuous in $s$. In other words $g(s)$ is a path. And for each fixed $s$ we have that $h(t)=F(s,t)$ is continuous in t. In other words $h(t)$ describes how the point $F(s,0)$ travels through time in a continuous path. These conditions or descriptions of $g(s)$ and $h(t)$ seem to capture the notion of "homotopic" to me. Is there some pathological reason why separately continuous is no good?