Let f 0 1 rarr r be a function the graph of f is the

Question

HONORS EXAMINATION IN REAL ANALYSIS, 2013

(1) Let f: [0, 1] → R be a function. The "graph" of f is the following subset of R²:

G = {(x, f(x)) | x ∈ [0, 1]}

Prove that f is continuous if and only if its graph is compact.

(2) Define f: R² → R as f(x, y) = √|xy| (the vertical lines here denote the absolute value). Is f is differentiable at the origin (0, 0)? Prove your answer.

(3) Among all 4-sided polygons in R² that enclose the origin (0, 0) and that have perimeter equal to 1, prove that there exists one of maximal area.

(4) Suppose that K₁ ⊃ K₂ ⊃ K₃ ⊃ · · · are infinitely many nested nonempty compact sets in Rⁿ. Prove that _i=1∩^∞ K_i (the intersection of all of these sets) is non-empty. Is the same result true for infinitely many nested nonempty compact sets in a general metric space?

(5) Let X and Y be metric spaces, with Y complete. Let A ⊂ X. Show that if f: A → Y is uniformly continuous, then f can be uniquely extended to a continuous function from the closure of A to Y, and that this extension is also uniformly continuous.

(6) State the Implicit Function Theorem and the Inverse Function Theorem. Choose one of these theorems to assume, and show how to prove the other as its corollary.

(7) Let r₁, r₂, r₃, ... be an enumeration of all of the rational numbers which lie in (0, 1). Let _i=1∑^∞ c_n be a convergent series whose terms are all positive real numbers. Define f: (0, 1) → R as follows:

f(x) = ∑_{{n|r_ncn.}

In other words, the sum is over all indices n for which r_n < x. Prove the following:

(a) f is increasing.

(b) f is continuous at each irrational number in its domain.

(c) f is discontinuous at each rational number in its domain.

(8) Define F: R³ → R⁴ as F(x, y, z) = (x² - y², xy, xz, yz). Let f denote the restriction of F to the unit-sphere S² ⊂ R³. Prove that the image f(S²) is a 2-dimensional manifold in R⁴. Is this image diffeomorphic to a familiar manifold? Hint: notice that f(-x, -y, -z) = f(x, y, z).

(9) If M is a compact orientable manifold whose boundary ∂M is nonempty, then there does not exist a smooth map f: M → ∂M whose restriction to ∂M equals the identity function. Prove this by providing the details of the following proof sketch: If such a function f exists, then,

0 ≠ ∫_∂Mω = ∫_∂Mf^∗(ω) = ∫_Mdf^∗(ω) = ∫_Mf^∗(dω) = 0

for appropriately chosen ω.

(10) Let λ ∈ R, and define f: R → R⁴ as f(t) = (cost, sin t, cos λt, sin λt). The image, f(R), inherits a topology from R⁴. Describe the closure of f(R). Under what condition on λ will f be a homeomorphism onto its image?