Suppose f (0, 1) → R is uniformly continuous. (a) Prove that f is bounded. (b) Prove that limx→0+ f(x) exists.

Let f:(b, ∞)→ℝ and let k∈ℝ. Prove that limx→∞k/f(x)=0 whenever limx→∞f(x)=∞.

Prove that the function |2x - f(x) = 32x3 6-4x does not tend to a limit as x tends to 3. Use the e- definition of continuity to prove that the function f(x) = x3 + x²-4x is continuous at the point c = 2. Prove that the function f(x) = x-2 2x+1 is uniformly continuous on the interval [1, 3], referring to any results from the module that you use.

Let g (0, 1] → R is continuous. Prove: (a) If g is uniformly continuous then then limx→0+ g(x) exists. (b) If limx→0+ g(x) = 3 then g is uniformly continuous.