= Solution
Let $i:Y\hookrightarrow M_f$ be the canonical inclusion and define
$$
r:M_f\longrightarrow Y,\qquad
r(y)=y,\quad r([t,x])=f(x).
$$
This is well-defined because $[0,x]=f(x)$ in the <mapping cylinder>, and continuity follows from the universal property of the <quotient topology>. Clearly $r\circ i=\operatorname{id}_Y$.
Define
$$
H_s(y)=y,\qquad
H_s([t,x])=[(1-s)t,x]
\qquad(0\le s\le1).
$$
The formula respects the gluing relation at $t=0$, so it descends from the disjoint union to a continuous <homotopy>. It satisfies $H_0=\operatorname{id}_{M_f}$ and $H_1=i\circ r$. Thus
$$
\boxed{i:Y\hookrightarrow M_f\text{ is a homotopy equivalence}},
$$
indeed $Y$ is a <deformation retraction>[deformation retract] of $M_f$.
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