Exercise 16, § 1

Let $X$ and $Y$ be topological spaces and let $f: X\rightarrow Y$ be a continuous mapping. Denote by $\varphi : C(X)\rightarrow$ Côn$(f)$ the canonical mapping. Prove that the spaces $Y/f(X)$ and Côn$(\varphi (C(X)))$ are homeomorphic.