What does it mean to say that a real-valued function on a metric space is uniformly continuous? Show that a continuous function on a closed interval in $\mathbb{R}$ is uniformly continuous.

What does it mean to say that a real-valued function on a metric space is Lipschitz? Show that if a function is Lipschitz then it is uniformly continuous.

Which of the following statements concerning continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ are true and which are false? Justify your answers.

(i) If $f$ is bounded then $f$ is uniformly continuous.

(ii) If $f$ is differentiable and $f^{\prime}$ is bounded, then $f$ is uniformly continuous.

(iii) There exists a sequence of uniformly continuous functions converging pointwise to $f$.