A subspace $Y$ of $\mathbb{R}$ is said to be a retract of $\mathbb{R}$ if there exists a continuous map $r : \mathbb{R} \longrightarrow Y$ such that $r(y) = y$ for every $y \in Y$.\\
(A) Show that $[0,1]$ is a retract of $\mathbb{R}$.\\
(B) Determine (with appropriate justification) whether every closed subset of $\mathbb{R}$ is a retract of $\mathbb{R}$.\\
(C) Show that $(0,1)$ is not a retract of $\mathbb{R}$.