1. Math
  2. Advanced Math
  3. 2 consider the set 234 i define the function g...

Question: 2 consider the set 234 i define the function g...

Question details

2. Consider the set 234 I define the function g by the equation g(r)- In this question you are going to study the integrability of the function g on [0, 1 Before you begin, sketch the graph of this function to understand it better. As always, make sure to justify all your answers. (a) What is the upper integral )? (b) Prove the following claim “For every positive integer n, and for every ε > 0, there erists a partition P of 0, 1 such that Lp(g) >1- -E. Hint: Before working on this question, answer the following extra questions in order. You do not need to submit these extra questions, but thinking about them will make answering 2b easier . Construct one partition P of0,such that Lp g) 0 Construct one partition P of [0, 1 with exactly 4 points (3 subintervals) such that Lp(g0.48 . Construct one partition P of [0, 1 with exactly 4 points (3 subintervals) such that Lp g 0.499 Construct one partition P of 0,1 with exactly 6 points (5 subintervals) such that LP(9)--0.001 (c) Prove the following claim: For every e > 0, there enists a partition P of [0,1 such that Lp(g) > 1-e. Hint: Use your answer to Question 2b. Mind your quantifiers and your proof structure (d) What is the lower integral /o (g)?

Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution