1. Let f : D → R be continuous at c ∈ D. Prove that there exists an M > 0 and a neighborhood U of c such that | f (x) | ≤ M for all x ∈ U ∩ D. 2. Complete the proof of Theorem 2.14 by showing that H ∩...

1. Let f : D → R be continuous at c ∈ D. Prove that there exists an M > 0 and a neighborhood U of c such that | f (x) | ≤ M for all x ∈ U ∩ D.

2. Complete the proof of Theorem 2.14 by showing that H ∩ D = f
^–1
(G ).

3. Let f :
 →
. Prove that f is continuous on R iff f^–1
(H ) is a closed set whenever H is a closed set.

Theorem 2.14

A function f : D → R is continuous on D iff for every open set G in 􀁜 there exists an open set H in
 such that H ∩ D = f^–1
(G ).