Hi everyone, I have a question from GR0568 and [edited away, please don't post actual exam problems]. Thank you!

GR0568_#62

Let K be a nonempty subset of Rn, where n>1. Which of the following statements must be true?

I. If K is compact, then every continuous real-valued function defined on K is bounded.

II. If every continuous real-valued function defined on K is bounded, then K is compact.

III. If K is compact, then K is connected.

(A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III

[answer is D. For II, we can give counterexample such as f(x)=sin(pi*x) where x is (-1,1), which f(x) is [-1,1]. However, the number of counterexample is limited if x belongs to an open set. So, I’m not sure that if every continuous real-valued function defined on K is bounded then will K must be a closed set?]