CalcNotes0208-page21 - (Section 2.8: Continuity) 2.8.21. 5....

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(Section 2.8: Continuity) 2.8.21. 5. Continuity on a set. A tricky thing to define! See “Continuity on a Set” by R. Bruce Crofoot, The College Mathematics Journal , Vol. 26, No. 1 (Jan. 1995) by the Mathematical Association of America (MAA). Also see Louis A. Talman, The Teacher’s Guide to Calculus (free online). Talman suggests: Let S be a subset of the domain of a function f ; i.e., S ± Dom f () . f is continuous on S ± ± a ² S , ± ² > 0, ± > 0 ± x ± S and x ² a < ³ ´ fx ² fa < µ · ¸ ¹ . • The definition essentially states that, for every element a in the set of interest, its function value is arbitrarily close to the function values of nearby x -values in the set. Note that we use instead of L , which we used to represent lim x ± a , because we need lim x ± a = (or possibly some one-sided variation) in order to have continuity on S . • This definition covers / subsumes our definitions of continuity on open intervals; closed intervals; half-open, half-closed intervals; and unions (collections) thereof.
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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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