1. (a) Let p ∈ and define f : → by f (x) = | x – p |. Prove that f is continuous. (b) Let S be a nonempty compact subset of 􀁜 and let p ∈ . Prove that S has a “closest” point to p. That is, prove...

1. (a) Let p ∈

and define f :

→

by f (x) = | x – p |. Prove that f is continuous.

(b) Let S be a nonempty compact subset of 􀁜 and let p ∈
. Prove that S has a “closest” point to p. That is, prove that there exists a point q in S such that | q – p | = inf {| x – p |: x ∈ S }.

2. Define f :

→

by f (x) = sin (1/x) if x ≠ 0 and f (0) = 0.

(a) Show that f is not continuous at 0.

(b) Show that f has the intermediate value property on
.