I had this question when I tried to find motivations behind the concept homotopy.
Suppose that $X$ and $Y$ are topological spaces and $H: X\times [0,1]\rightarrow Y$ is a continuous map between the product space and $Y$. Fix $r\in [0,1]$, I define $f: X\rightarrow Y$ to be $f(x) = H(x,r)$, is the induced function $f$ a continuous function between $X$ and $Y$?
I tried to solve it. Take a set $O$ open in $Y$, $f^{-1}(O)$ is the $x$ part of $H^{-1}(O)\cap X\times\{0\}$, but I didn't know how to proceed. I think the answer to my question might be $f$ is not necessarily a continuous function.
Any help is appreciated!