For $ E \subset \mathbb{R}$, consider the following statements:
$P$: Every continuous function $f: E \to \mathbb{R}$ is uniformly continuous.
$Q$: $E$ is compact.
$R$: Every continuous function $f: E \to \mathbb{R}$ is bounded.
Which of the following is(are) false?
(A) $ R \nRightarrow P$
(B) $ P \implies Q$
(C) $ Q \implies P$
(D) $ R \implies Q$
Solution:
$ R \Rightarrow Q$ is correct statement, it's a theorem. Similarly, $Q \Rightarrow P $ is also correct. That means, options (C),(D) are correct.
Please help me in verifying options (A) and (B), how should I approach these two options? In general, how to get a counter-example for such questions. Thanks in advance.
$R$ does imply $P$ because $R$ implies $Q$ and $Q$ implies $P$.
(B) is false: consider $\mathbb N$ with the usual metric. Any real function on this space is uniformly continuous but this space is not compact.