HONORS EXAM 2012 REAL ANALYSIS
1. Prove or disprove: If {pn}∞n=1 is a sequence of polynomials and ∑pn → f uniformly on R as n → ∞, then f is a polynomial.
2. Let C = {f: [0, 1] → [0, 1] | f is continuous}, the set of continuous maps from the interval [0, 1] to itself. Define a metric d on C by d(f, g) = maxx∈[0,1]|f(x) - g(x)|. Let Ci and Cs be the sets of injective and surjective elements, respectively, of C. Prove or disprove the following:
(a) Ci is closed in C.
(b) Cs is closed in C.
(c) C is connected.
(d) C is compact.
3. Define a sequence of functions f1, f2, . . . :[0, ∞) → R by fn(x) = sin(x/n)/x +(1/n). Discuss the convergence of {fn} and {f'n} as n → ∞.
4. Recall the Intermediate Value Theorem:
Let f: [a, b] → R be a continuous function, and y any number between f(a) and f(b) inclusive. Then there exists a point c ∈ [a, b] with f(c) = y.
(a) Prove the Intermediate Value Theorem.
(b) Prove or disprove the following converse to the Intermediate Value Theorem:
If for any two points a < b and any number y between f(a) and f(b) inclusive, there is a point c ∈ [a, b] such that f(c) = y, then f is continuous.
(c) Prove or disprove the following fixed-point theorem:
Let g: [0, 1] → [0, 1] be continuous. Then there exists a fixed point x ∈ [0, 1] (that is, a point x such that g(x) = x).
5. This question deals with the Riemann integral.
(a) Let S be the unit square [0, 1] × [0, 1]. Define f: S → R by setting
For each of the following integrals, compute its value or show that it does not exist:
0∫1(0∫1f(x, y) dx)dy, 0∫1(0∫1f(x, y) dy)dx, ∫Sf(x, y).
(b) What conditions on f would guarantee that all three integrals exist and are equal?
(c) Give an example of a function g and a domain D such that ∫D|g| exists but ∫Dg does not.
6. Let f: R2 → R2 be smooth (C∞) and suppose that
∂f1/∂x = ∂f2/∂y, ∂f1/∂y = - (∂f2/∂x).
(These are the Cauchy-Riemann equations, which arise naturally in complex analysis.)
(a) Show that Df(x, y) = 0 if and only if Df(x, y) is singular, and hence f has a local inverse if Df(x, y) ≠ 0. Show that the inverse function also satisfies the Cauchy-Riemann equations.
(b) Give an example showing that the statement in part (a) (f has a local inverse if Df(x, y) ≠ 0) may be false if f does not satisfy the Cauchy-Riemann equations.
7. Let M be a compact 2-manifold in R2, oriented naturally; give the boundary ∂M the induced orientation. Let f: R2 → R be a smooth (C∞) function such that f(x) = 0 for any x ∈ ∂M.
(a) Prove that
∫Mf · (∂2f/∂x2 + ∂2f/∂y2) dx ∧ dy = -∫M((∂f/∂x)2 + (∂f/∂y )2) dx ∧ dy.
(b) Deduce from (a) that if, in addition, f is harmonic on M (that is, ∂2f/∂x2 + ∂2f/∂y2 = 0 on M), then f(x) = 0 for any x ∈ M.
8. (a) (i) Let f be the polar coordinate map given by (x, y) = f(r, θ) = (r cos θ, r sin θ). Compute f∗(dx), f∗(dy), and f∗(dx ∧ dy).
(ii) Compute ∫C xy dx, where C = {(x, y)| x2 + y2 = 1, x ≥ 0, y ≥ 0}, the portion of the unit circle in the first quadrant, oriented counter-clockwise.
(b) Let M be a manifold, possibly with boundary. A retraction of M onto a subset A is a smooth (C∞) map φ: M → A such that φ(x) = x for all x ∈ A. (For example, the map φ(x) = x/||x|| is a retraction of the punctured plane R2\{(0, 0)} onto the unit circle S1.) Prove the following theorem:
There does not exist a retraction from the plane R2 onto the unit circle S1.
(Hint: Consider the 1-form x dy - y dx/x2 + y2, which is defined in an open set containing S1.)
9. (a) Let ω1 and ω2 be differential forms defined on the same domain.
(i) If ω1 and ω2 are closed, must ω1 ∧ ω2 also be closed? If ω1 ∧ ω2 is closed, must ω1 and ω2 also be closed?
(ii) If ω1 and ω2 are exact, must ω1 ∧ ω2 also be exact? If ω1 ∧ ω2 is exact, must ω1 and ω2 also be exact?
(b) Show that every closed 1-form on the punctured space R3\{(0, 0, 0)} is exact.