Let f : D → and let c ∈ D. We say that f is bounded on a neighborhood of c if there exists a neighborhood U of c and a number M such that | f (x) | ≤ M for all x ∈ U ∩ D. (a) Suppose that f is...

Let f : D →

and let c ∈ D. We say that f is bounded on a neighborhood of c if there exists a neighborhood U of c and a number M such that | f (x) | ≤ M for all x ∈ U ∩ D.

(a) Suppose that f is bounded on a neighborhood of each x in D and that D is compact. Prove that f is bounded on D.

(b) Suppose that f is bounded on a neighborhood of each x in D, but that D is not compact. Show that f is not necessarily bounded on D, even when f is continuous.

(c) Suppose that f : [a, b] →

has a limit at each x in [a, b]. Prove that f is bounded on [a, b].