CalcNotes0208-page20

CalcNotes0208-page20 - (Section 2.8: Continuity) 2.8.20...

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(Section 2.8: Continuity) 2.8.20 FOOTNOTES 1. A function with domain ± that is only continuous at 0. (Revisiting Footnote 1 in Section 2.1.) Consider the following function f . fx () = 0, if x is a rational value x ,i f x is an irrational value ± ² ³ f is continuous at 0, because f 0 = 0 , and we can use the Squeeze (Sandwich) Theorem to prove that lim x ± 0 = 0 , also. The discontinuities at the nonzero real numbers are neither removable, jump, nor infinite. 2. Infinite discontinuities: alternate definition. Some textbooks only require that ±² or ³² as x ± a + or x ± a ² . Such a definition would allow us to say that the following function has an infinite discontinuity at 0. = 1 x f x is a rational value ± 1 x f x is an irrational value ² ³ ´ ´ µ ´ ´ If we adopt the one-point compactification of the real numbers, also known as the real
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This note was uploaded on 12/29/2011 for the course MATH 150 taught by Professor Sturst during the Fall '10 term at SUNY Stony Brook.

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