|Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:||(67)||
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Solve the following 5 problems. Each is worth 20 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 50 minutes.
Allowed Material: Any calculating device that is not capable of displaying text.
(In the real exam each of the questions will appear on a separate page and there will be several blank pages stapled with your exam booklet. You will have space for your name and student number at the top of each page.)
Problem 2. We say that a function is an inverse of a function if , where is the identity function, defined by for all . Show that a function has an inverse if and only if the following two conditions are satisfied:
Problem 3. Sketch, to the best of your understanding, the graph of the function
Problem 4. Suppose that is, for each natural number , some finite set of numbers and that and have no members in common if . Define as follows:
Problem 5. Suppose that satisfies for all and and that is continuous at 0. Prove that is continuous everywhere.
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