I'm having trouble with part of the definition of homotopy, but I may just have a misunderstanding about continuous maps from product spaces. This is the definition my book uses:
Two continuous function $f_0,f_1:X\to Y$, are said to be homotopic if there exists a continuous map $F:X\times I\to Y$ such that $F(x,0)=f_0(x)$ and $F(x,1)=f_1(x)$. $F$ is called a homotopy between $f_0$ and $f_1$ and for each $t\in [0,1]$, we denote $F(x,t)$ by $f_t(x)$, whence $f_t:X\to Y$ is a continuous map.
I don't understand why the bold part is true. Why does $F$ continuous imply each $f_t$ is continuous? Is it true in general for a continuous map $H:X\times Y\to Z$, the maps fixing $y$, $H(X,y)$ or $x$, $H(x,Y)$ are continuous maps into $Z$? I would think yes since if we give $X\times\{y\}$ the subspace topology, the restriction of the continuous map $H$ to $X\times\{y\}$ is also continuous. Can anyone please clarify?
If $y \in Y$ is a point, then the map $i_y: X \to X \times Y$, given by $i_y(x)=(x,y)$, is continuous by definition of the product topology on $X \times Y$. Then if $H: X \times Y \to Z$ is a continuous map, then $H_y: X \to Z$, given by $H_y(x) = H(x,y)$ is continuous, because it's the composition of two continuous maps $H_y = H \circ i_y$. Now specialize to $Y = [0,1]$.