Define the function f: R rightarrow R by f (x) = x + 1 if x 0 and f (0 ) = 4. Sh

Question

Define the function f: R rightarrow R by f (x) = x + 1 if x 0 and f (0 ) = 4. Show that limx rightarrow 0 f (x) = 1 and f is not continuous at x = 0. Find the following limits or determine that they do not exist: Let D be the set of real numbers consisting of the single number x0. Show that the set D has no limit points. Also show that the set N of natural numbers has no limit points. Let D be a nonempty subset of R that is bounded above. Is the supremum of D a limit point of D? Explain why, in the definition of lim x rightarrow x0 f (x), it is necessary to require that x0 be a limit point of D. A point x0 in D is said to be an isolated point of D provided that there is an r > 0 such that the only point of D in the interval (x0 - r, x0 + r) is x0 itself. Prove that a point x0 in D is either an isolated point or a limit point of D.

Explanation / Answer

For #6, the answer is yes.

Because if the supremum x was NOT a limit point of D, then there is a neighborhood around x that does not contain an element in D meaning there is another number between x and D meaning that x could not have been a supremum.

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