I have this exercise:
If Y is path-connected, show that there is only one homotopy-class of continuous functions from $[0,1]$ to Y.
What I need to show is that if I have two continuous functions $f_1,f_2: [0,1]\rightarrow Y$ they are homotopic. I must find an F such that $f: I \times I\rightarrow Y$, so that F is continuous, and $F(t,0)=f_1(t), F(t,1)=f_2(t)$.
There is one way that seem very natural to construct F here, that is for every t, since Y is path-connected, there is a path from $f_1(t)$ to $f_2(t)$, this path can be written: $f_t(s): [0,1]\rightarrow Y$, where $f_t(0)=f_1(t), f_t(1)=f_2(t)$.
Then we just denote $F(t,s)=f_t(s)$.
By construction this will be a homotopy(not path-homotopy) between $f_1, f_2$ if we have that F is continuous. But how do I show that it is continuous? Or is it even continuous? What I know about continuoity is this:
$F(t,0), F(t,1)$ is continuous in t. $F(t,s)$ is continuous in s for all t. But still it is not enough, we need joint continuity. Any tips on how to show continuity?