 Solution

Every map has $f(x)=f(x)$, so the relation is reflexive. Equality is symmetric, so $f(x)=f(y)$ for all the specified maps is equivalent to $f(y)=f(x)$; the relation is symmetric. If $x\mathcal{R}y$ and $y\mathcal{R}z$, every continuous map to every Hausdorff space has $f(x)=f(y)=f(z)$, proving $x\mathcal{R}z$. Thus $\mathcal{R}$ is an equivalence relation.