Connectedness
Exercise 2.1
Suppose that $X \subseteq \mathbb{R}^n$ is open and connected. Show that $X$ is path-connected. (Thus, for open subsets of Euclidean space, connectedness and path-connectedness are equivalent.)
Continuity
Exercise 2.2
Let $f: X \to Y$ be a function between metric spaces $X$ and $Y$ and suppose that $X$ is compact. Prove that $f$ is continuous if and only if its graph $G_f = \set{(x, f(x)) : x \in X}$ is compact.