1. Let f : D → and let c be an accumulation point of D. Suppose that limx → c f (x) > 0. Prove that there exists a deleted neighborhood U of c such that f (x) > 0 for all x ∈ U ∩ D. 2. Define f : → ...

1. Let f : D →

and let c be an accumulation point of D. Suppose that lim_{x → c}
f (x) > 0. Prove that there exists a deleted neighborhood U of c such that f (x) > 0 for all x ∈ U ∩ D.

2. Define f :

→

by f (x) = x if x is rational and f (x) = 0 if x is irrational. Prove that f has a limit at c iff c = 0.

3. Let f : D →

and let c be an accumulation point of D. Suppose that f has a limit at c. Prove that f is bounded on a neighborhood of c. That is, prove that there exist a neighborhood U of c and a real number M such that | f (x)| ≤ M for all x ∈ U ∩ D