# Question

a. Show that an unbounded set cannot have content 0.

## Answer to relevant Questions

a. If C is a set of content 0, show that the boundary of C also has content 0. b. Give an example of a bounded set C of measure 0 such that the boundary of C does not have measure 0.Show that if C has content 0, then C C A for some closed rectangle A and C is Jordan-measurable and ∫ AXC = 0.Let C C A x B be a set of content 0. Let A1 C A be the set of all x Є A such that {y Є B: (x, y) Є C} is not of content 0. Show that A1 is a set of measure 0.Let f: [a, b] x [c, d] → R be continuous and suppose D 2 f is continuous. Define f (y) = ∫ ba f (x, y) dx. Prove Leibnitz' Rule: f1 (y) = ∫ ba D2 f (x, y) dx.Find a counter-example to Theorem 5-2 if condition (3) is omitted. Following the hint, consider f: (- 2π, 2π) →R2 defined byPost your question

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