1. (a) Let f : D → and define | f | : D → by | f | (x) = | f (x) |. Suppose that f is continuous at c ∈ D. Prove that | f | is continuous at c. (b) If | f | is continuous at c, does it follow that f...

1. (a) Let f : D →

and define | f | : D →

by | f | (x) = | f (x) |. Suppose that f is continuous at c ∈ D. Prove that | f | is continuous at c.

(b) If | f | is continuous at c, does it follow that f is continuous at c? Justify your answer.

2. Let f : D →

be continuous at c ∈ D and suppose that f (c) > 0. Prove that there exists an α > 0 and a neighborhood U of c such that f (x) > α for all x ∈ U ∩ D.