Let $X$ and $Y$ be topological spaces. Assume $Y$ is contractible (hence, path- connected).

Let $f,g: X \to Y$ be continuous maps. At any fixed $x\in X$, there is a path $P_x: [0,1]\to Y$ from $f(x)$ to $g(x)\in Y$ such that $P_x(0)=f(x)$ and $P_x(1)=g(x)$

We define $F: X\times [0,1]\to Y$ by $F(x,t) = P_x(t)$. Now, $$F(x,0)= P_x(0)=f(x), F(x,1)= P_x(1)=g(x)$$ for any $x\in X$ (This is why $F$ is homotopy-like). Clearly, $F$ is continuous at time $0$ and at time $1$. Is $F$ necessarily continuous at any time $t$?

Thanks in advance for any help.