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Due Friday, October 31 (Happy Halloween!)
- Sect. 3.7:
- For any set D and any number a, let [D < a] denote the set of all elements of D which are less than a, and define [D > a] analogously.
Suppose that a is a limit point of both [D < a] and [D > a]. Let f be a real-valued function on D. Denote by f<a the restriction of f to [D < a], and define f>a analogously.
- Prove that if a is a limit point of either [D < a] or [D > a], then a is a limit point of D.
- Explain why all the relevant domains ( [D < a], etc.) are nonempty.
- Let L be a real number. Show that if f has the limit L at a, then both f<a and f>a have the limit L at a.
(Note: “f has the limit L at a” means the same as “the limit as x approaches a of f(x) is L” )- State and prove the converse of the preceding statement.
NOTE: As stated on the Course Home Page, all due dates are tentative. Assignments, or parts of assignments, may be postponed to a later date.
Last modified Oct 23, 2008
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