 Solution

Let $i:Y\hookrightarrow M_f$ be the canonical inclusion and defineThis is well-defined because $[0,x]=f(x)$ in the mapping cylinder, and continuity follows from the universal property of the quotient topology. Clearly $r\circ i={\mathrm{id}}_Y$.

DefineThe formula respects the gluing relation at $t=0$, so it descends from the disjoint union to a continuous homotopy. It satisfies $H_0={\mathrm{id}}_{M_f}$ and $H_1=i\circ r$. Thusindeed $Y$ is a deformation retract of $M_f$.