Question1. I can prove that Every metric space is Hausdorff. Where do they use this fact in this proof?
Question2. Let $C$ be a closed subset of $X$ and $p\in X\setminus C$. How do I prove the function $f:X\to I=[0,1]$ by $f(x)=\min\{\frac{d(x,C)}{d(p,C)},1\}$. How do I prove that $f$ is continuous?
My attempt:- $g(x)=d(x,C)$ is a continuous function. Since, for $c\in C$, $d(x,c)\leq d(x,y)+d(y,c)\implies d(x,C)-d(x,y)\leq d(y,c)\implies d(x,C)-d(x,y)\leq d(y,C) \implies d(x,C)-d(y,C)\leq d(x,y) $ . From this inequality we can deduce that $g$ is continuous. $d(p,C)\neq 0$ is constant. So,$\frac{d(x,C)}{d(p,C)}$ is continuous. The minimum of two continuous real-valued functions $u,v$ is continuous, since $\min \{u(x),v(x)\}=\frac{1}{2}(u(x)+v(x)-|u(x)-v(x)|)$. Hence, $f$ is continuous. Am I correct?
Your proof is correct. I do not know which definition of "completely regular" is used in your textbook. If it means that any closed $C$ and any $x \notin C$ can be separated by a continuous function, then "Hausdorff" is irrelevant. However, many authors define "completely regular = Hausdorff + functional separation".