Honors Exam in Real Analysis 2007
1. Suppose K is a subset of Rn. Prove that K is compact if and only if every continuous function f: K → R is bounded.
2. Suppose f: (0, 1] → R is differentiable and satisfies |f'(x)| < 1 there. Show that the sequence {f(1/n)} converges.
3. Suppose f: [0,∞) → R is of class C2, and f(x) → 0 as x → ∞.
(a) If f'(x) → b as x → ∞, show that b = 0.
(b) If f'' is bounded, show that f'(x) → 0 as x → ∞.
(c) Give an example of such an f for which f'(x) does not converge as x → ∞.
4. Suppose f: [0, 1] → R is upper semicontinuous: This means that for every x ∈ [0, 1] and every ε > 0, there exists δ > 0 such that |y -x| < δ implies f(y) < f(x) + ε. Prove that f is bounded above and achieves its maximum value at some x ∈ [0, 1].
5. Define a function f: [0, 1] → R by setting f(x) = 1/n if the first 0 after the decimal point in the decimal expansion of f occurs in the nth place after the decimal point, and f(x) = 0 if there are no 0's after the decimal point. Prove that f is Riemann-integrable. (To avoid ambiguity, choose decimal representations ending in 9's instead of 0's when both are possible. For example,
f(1/10) = f(0.0999 ... ) = 1,
f(1/2) = f(0.4999 ... ) = 0,
f(1) = f(0.9999 ... ) = 0.
Note that only digits after the decimal point are counted.)
6. Suppose f: Rn → Rk is continuous. Let λ be a positive real number, and assume that for every x ∈ Rn and a > 0, f(ax) = aλf(x).
(a) If λ > 1, show that f is differentiable at 0.
(b) If 0 <λ< 1, show that f is not differentiable at 0.
(c) If λ = 1, show that f is differentiable at 0 if and only if it is linear.
7. Let M2 be the set of 2 × 2 real matrices, identified with R4 in the obvious way, and define F: M2 → M2 by F(X) = X2 (i.e., the matrix X multiplied by itself). Does F have a local smooth inverse in a neighborhood of I (the 2 × 2 identity matrix)? Answer the same question when I is replaced by
Prove your answers correct.
8. Let T ⊂ R3 denote the doughnut-shaped surface obtained by revolving the circle (y - 2)2 + z2 = 1 around the z-axis. Give T the orientation determined by the outward unit normal.
(a) Compute the surface area of T.
(b) Compute the integral ∫Tω, where ω is the 2-form z dx ∧ dy.
9. Suppose M is a smooth, compact n-manifold with boundary in Rn. If f is a smooth real-valued function and X is a smooth vector field on Rn, use the general version of Stokes's theorem to prove the following "integration by parts formula":
∫M(grad f,X) dV = ∫∂Mf(X, N) dV - ∫Mf div X dV,
where (·, ·) denotes the Euclidean inner product or dot product, and N denotes the outward unit normal to ∂M. Explain what this has to do with integration by parts.