Connectedness

• When Does a Connected Set Meet the Boundary of a Set?
  A connected set intersects the boundary of A if it meets both A and its complement; the proof follows by separating its interior portions.
• Does a Connected Graph Imply That a Function Is Continuous?
  A function can be discontinuous even when its graph is connected. The oscillating function sin(1/x), extended by f(0)=0, is a concrete example.
• Are Singleton Sets in ℝ Both Open and Closed?
  In ℝ with the usual topology, a singleton {a} is closed but not open. Proofs of both facts, and why only ∅ and ℝ are both open and closed.