Comprehensive Examination Advanced Calculus June 1999

A passing mark will be given if 5 problems are completely solved.

1. Use the open cover definition of compactness to prove that if a subset A of the real numbers is compact and infinite, then it has an accumulation point.

2. Let S be a non-empty, bounded subset of ℝ. Let k ∈ ℝ and define kS = {ks : s ∈ S}. (a) Prove that if k > 0, sup kS = k sup S.

(b) State and prove similar results for sup kS and k ≤ 0.

3. Suppose that f : [a, b] → ℝ.

(a) Define f is Riemann integrable.

(b) Give an example of a bounded function on [0,1] which is not Riemann integrable and prove by using the definition that it is not.

4. Let I = [0, 1]. Prove each of the following.

(a) ∑_{n=1}^{∞} x^{n}(1 − x) converges pointwise on I but not uniformly.

(b) ∑_{n=1}^{∞}(−1)^{n}x^{n}(1 − x) converges uniformly on I.

5. Suppose that a < b < c, f is continuous on [a, c] and f′′(x) exists for all x ∈ (a, c). If f(a) = a, f(b) = b, and f(c) = c, prove that there exists some d ∈ (a, c) such that f′′(d) = 0.

6. Prove that if f : ℝ^{2} → ℝ is differentiable, then f is not injective.

7. Suppose that f(x, y) is a real-valued function on {(x, y) : a ≤ x ≤ b, A ≤ y ≤ B}. For each step of the following argument, give a reasonable condition on f which makes the step valid and justify the step. Define

g(x) = ∫_{A}^{B} f(x, y) dy.

= ∫_{A}^{B} [f(a, y) + ∫_{a}^{x} \frac{∂f}{∂x}(t, y)dt] dy Step 1

= ∫_{A}^{B} f(a, y) dy + ∫_{A}^{B} ∫_{a}^{x} \frac{∂f}{∂x}(t, y)dt dy Step 2

= ∫_{A}^{B} f(a, y) dy + ∫_{a}^{x} ∫_{A}^{B} \frac{∂f}{∂x}(t, y)dy dt Step 3

Step 4  Therefore g′(x) = ∫_{A}^{B} \frac{∂f}{∂x}(x, y)dy