If $f$ and $g$ are continuous maps from $X$ to $Y$, then a homotopy $H$ between $f$ and $g$ is a continuous map from $X \times [0,1]$ to $Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for each $x \in X$.
I know that we want a homotopy between $f$ and $g$ to be a "continuous deformation" where $H(x,t_0)$ is a continuous map for each fixed $t_0 \in [0,1]$, and also the deformation takes places continuously. However, I still do not understand why we wnat to require a homotopy $H$ to be continuous on the product space $X \times [0,1]$.
Instead, I would think we can require a homotopy $H$ to be continuous in each variable separately, meaning for each $t_0 \in [0,1]$ the map $H(x,t_0)$ is continuous on $X$ and for each $x_0 \in X$, the map $H(x_0,t)$ is continuous on $[0,1]$.
I know that if a function is continuous on $X\times Y$, then $f$ is continuous in each variable separately, but not vice versa. When I think of a homotopy as a continuous deformation, the notion of separate continuity makes more sense to me. Why do we want a homotopy to be continuous on $X \times [0,1]$?