|Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I:||(51)||
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Solve the following 4 problems. Each is worth 25 points although they may have unequal difficulty. Write your answers in the space below the problems and on the front sides of the extra pages; use the back of the pages for scratch paper. Only work appearing on the front side of pages will be graded. Write your name and student number on each page. If you need more paper please ask the tutors. You have an hour and 45 minutes.
Allowed Material: Any calculating device that is not capable of displaying text.
Problem 1. Let and be continuous functions defined for all , and assume that . Define a new function by
Problem 2. We say that a function is locally bounded on some interval if for every there is an so that is bounded on . Prove that if a function (continuous or not) is locally bounded on a closed interval then it is bounded (in the ordinary sense) on that interval.
Hint. Consider the set is bounded on and think about P13.
Problem 4. Draw a detailed graph of the function