Trouble with an homotopy

Question

I am trying to prove that an injective map $f$ from the circle to a disk with a point removed, $Y$, is homotopic to a map that sends the circle to a point, $g$. In order to do that, I have defined an homotopy from $f$ to the constant map $f(1)$ in this way: \begin{align*} H\colon \mathit{S}^{1} \times I \mapsto Y\\ (e^{2\pi si},t) \mapsto f(e^{2\pi (1-t)si}) \end{align*} $H$ is an homotopy. After that, since $Y$ is path-connected, the constant map $f(1)$ and any constant map are going to be homotopic. Therefore, because of the transitivity, I obtain that $f$ ang $g$ are homotopic. I haven't use the fact that $f$ is injective. Can someone tell me please where is my argument wrong?