Let $C(X,Y)$ the set of continuous functions from $X$ to $Y$. Consider in $C(X,Y)$ the compact-open topology. If $f$ is homotopic to $g$ then there is path from $f$ to $g$. Suppose that the homotopy between $f$ and $g$ is $F:X\times [0,1]\to Y$. A candidate for the path joining $f$ with $g$ is $\lambda:[0,1]\to C(X,Y);t\mapsto f_t$, where $f_t(x)=F(x,t)$. I'm having trouble showing that $\lambda$ is continuous...