We know the definition of homotopy equivalence:
Let $f,g:X \to Y$ be continuous between topological spaces.
We say $f$ is homotopic to $g$ if there is a continuous map $H : X \times I \to Y \ \ $ satisying
$H(x,0) = f(x)\ and \ H(x,1) = g(x)$.
Now, if $\gamma:I \to X $ is a closed curve such that $\gamma(0) = p =\gamma(1)$ for some $\gamma \in X$.
If we define $H : I \times I \to Y \ $ by $H(s,t) = \gamma (st)$ , then $H(s,1) = \gamma(s)\ and \ H(s,0) = \gamma(0) = p$
So we have that any closed curve is homotopic to a constant map, but I know that is not true! For example not all closed curve can be contract to a point on a torus.
So, where is the main point? is $H(s,t) = \gamma (st)$ dis-continuous ? or I misunderstand the definition between homotopy and homotopy equivalence?