Show that if $f,g:X \rightarrow CY$ are continuous, then they are homotopic

Question

$X,Y$ are topological spaces and $CY=Y\times[0,1]/(x,1)\tilde{}(y,1)$ for all $x,y\in Y$ the cone of $Y$. In order for $f,g$ to be homotopic I have to find a homotopy $H(x,t):X\times [0,1] \rightarrow CY$ with $H(x,0)=f(x)$ and $H(x,1)=g(x)$.

Since $H$ is a map to the cone doesn't $H(x,1)$ has to be constant? If yes, how can I go on? Thanks for your help.

Of course not. Only if you're giving a homotopy to a constant map will $H(x,1)$ be constant. — Ted Shifrin, May 09 '17 at 20:25
Now I See that$ H(x,1) $ does not have to be constant. However, I Do not know how to proceed to get a homotopy between both functions — deavor, May 09 '17 at 20:32

score 1 · Accepted Answer · answered May 10 '17 at 16:50

Hint: First consider the special case where $X=CY$, $f$ is identity map, $g$ is constant map to the tip point of the cone. Try to construct a homotopy. This construction should be straightforward. Once this is done try to do the original problem.

(If you have further doubts about the technical problem of the continuity of the construction, see this MSE answer)

Show that if $f,g:X \rightarrow CY$ are continuous, then they are homotopic

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