I was trying to show the two following things: 1) any two maps from a topological space $X$ to $I=[0,1]$ are homotopic, 2) any two maps from $I=[0,1]$ to a path connected set $Y$ are homotopic.
For the second I thought I could do it like this:
Let $f:I\rightarrow Y$ en $g:I\rightarrow Y$ continuous maps. Now define a function $F:I\times I\rightarrow Y$ such that $F(x,t)=(1-t)f((1-t) x) + t g(tx)$. Then we see that $F(x,0)=f((1-0) x) + 0\cdot g(0\cdot x)=f(x)$ and $F(x,1)=(1-1)f((1-1) x)+1\cdot g(1\cdot x)=f(x)$. Furthermore, $F$ is continuous since it's a composition of continuous functions. Now we can conclude that $f$ and $g$ are homotopic.
However, I don't use the fact that $Y$ is path connected, so I think this can't be right. I already found this, however, I don't understand what the $e_{f(0)}$ and $e_{g(0)}$ are. Furthermore, I found this, but in the answers they don't create the $F$-function.
Could someone help me find the right $F$-function to show this? And also for the first part?
For the first question it is enough to show that the map $f$ is homotopic to the constant map $g(x)=0$, define $H_t(x)=tf(x)$.
For the second, you can also show that any map from $I\rightarrow Y$ is homotopic to $g(t)=y_0, y_0\in Y$. Firstly $f$ is homotopic to $g_0(x)=f(0)$ by using $H_t(x)=f(tx)$, consider a path $c:[0,1]\rightarrow Y$ such that $c(0)=f(0), c(1)=y_0$, write $H'_t(x)=c(t)$, $H'$ defines an homotopy between $g$ and $g_0$.