Math 554/703I - Analysis I
Test 3 Sample Problems
April 14, 2004
Here are 10 problems to provide examples from all areas covered since Test 2. The
test on Wednesday will have 7-8 problems.
Directions: To receive credit, you must justify your statements unless otherwise
stated. Answers should be provided in complete sentences.
1. Give an example of each of the following and (very) briefly justify your answer:
(a) A bounded set of real numbers that is not compact.
(b) A connected set of real numbers that is not compact.
(c) A real-valued continuous function that does not satisfy the Extreme Value Theorem.
(d) A real-valued continuous function that does not satisfy the Intermediate Value Theorem.
2. a.) Spell ‘Bolzanno-Weierstrass’.
b.) State the Bolzanno-Weierstrass property for a subset of real numbers.
3. a.) Define open cover for a set.
b.) Define what it means for a set to be compact.
c.) Suppose Kis compact and f:K→IR is continuous. Prove that f[K] is compact.
4. State and sketch a proof of the Heine-Borel theorem.
5. Prove that each closed and bounded subset of real numbers is a compact set.
6. a.) Define what it means for a set Ato be a connected subset of real numbers.
b.) Show that a subset Aof real numbers is connected implies that A is an interval.
7. State and prove the Intermediate Value Theorem
8. a.) Define what it means for fto be differentiable at x=x0.
b.) Prove that fis continuous at x=x0if fis differentiable at x=x0.
9. Show that the function f(x) = x5is one-to-one from [0,∞)onto [0,∞). Prove that its
inverse is continuous.
10. Pick one of the following:
a.) State and prove the Chain Rule.
b.) State and prove the Quotient Rule