Let $f,g$ be two continuous functions between topological spaces $X$ and $Y$. Suppose $f$ and $g$ are homotopic, can we say $f(X)$ and $g(X)$ are homotopic?
I think intuitively they are. Because $f\simeq g$ means there exists an $H:X\times I\rightarrow Y$ which is continuous such that $H(x,0)=f(x),H(x,1)=g(x)$. I think it is saying that the image of $f$, namely $f(X)$, can continuously deform into $g(X)$. Then it seems to mean $f(X)$ and $g(X)$ are homotopic, by the geometric intuition of homotopy.
But trouble comes when I try to prove it. For this end, I need to construct a map from $f(X)$ to $g(X)$ and a map from $g(X)$ to $f(X)$. However, after some try, I cannot find such map.
Can anyone help?