Masters Comprehensive Exam

Advanced Calculus

June 1992

Justify all work. Four correct problems (not parts) will constitute a pass.

1.a. Define the terms countable and uncountable set. b. Let Q([0, 1]) denote the set of functions on [0,1] with range contained in the

rational numbers. Prove or disprove: Q([0, 1]) is uncountable. c. Prove or disprove: The set of continuous functions in Q([0, 1]) is uncountable. d. Prove that there is a sequence of continuous functions (f_{n})_{n=1}^{∞} on [0,1] such

that for every continuous function g on [0,1] there is a subsequence of (f_{n})_{n=1}^{∞} which converges to g uniformly. (You may use a certain famous theorem due to Weierstrass.)

2.a. State Taylor’s Theorem with (any correct form of the) remainder.

Let f(x) = \{ \begin{matrix}exp(−\frac{1}{x^{2}} ) & if x ≠ 0 \\ 0 & if x = 0 \end{matrix} .

b. Show that f′(0) = f′′(0) = 0.

Actually f^{(n)}(0) = 0 for all n. You may use this fact in the next two parts.

c. Determine the set of all real numbers x_{0} such that f has a power series repre-

sentation (with positive radius of convergence) of the form ∑_{n=0}^{∞} a_{n}(x − x_{0})^{n}. d. Show that there is a non-zero C^{∞}-function g on ℝ such that g(x) = 0 for all

x ≤ 0.

3. Suppose that D is the unit square in ℝ^{2}, i.e., D = {(x, y) : 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}.

a. State Fubini’s Theorem for the case of continuous functions f(x, y) on D. b. Let D′ = {(x, y, 0) : 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}. With D identified with D′ in

the obvious way and f(x, y) as above, consider the surface integral of f over D′. In this context what does the equation dy dx = −dx dy mean?

c. Explain the apparent contradiction between Fubini’s Theorem and the equation in Part b.

4. Consider the series

∑_{n=1}^{∞} \frac{(−1)^{n}}{\sqrt{n}} sin(1 + \frac{x}{n}).

a. Show that the series converges uniformly on any finite subinterval of ℝ^{+}. b. Does the series converge uniformly on ℝ^{+}?

5. Suppose that (f_{n}) is a sequence of continuous functions on [0,1] such that f_{n}(x) ≤ f_{n+1}(x) for all x ∈ [0, 1] and n ∈ ℕ and lim_{n→∞} f_{n}(x) = g(x) < +∞ for all x ∈ [0, 1].

a. Show that there is an x_{0} ∈ [0, 1] such that g(x_{0}) = inf{g(x) : x ∈ [0, 1]}. b. Show that the corresponding result for sup{g(x) : x ∈ [0, 1]} fails.

Typeset by AMS-TEX