HONORS EXAM 2012 REAL ANALYSIS

1. Prove or disprove: If {pn}^∞_n=1 is a sequence of polynomials and ∑p_n → 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) = max_x∈[0,1]|f(x) - g(x)|. Let C_i and C_s be the sets of injective and surjective elements, respectively, of C. Prove or disprove the following:

(a) C_i is closed in C.

(b) C_s is closed in C.

(c) C is connected.

(d) C is compact.

3. Define a sequence of functions f₁, f₂, . . . :[0, ∞) → R by f_n(x) = sin(x/n)/x +(1/n). Discuss the convergence of {f_n} 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.