 T is continuous at x if and only if, for every e > 0, there is a <5 > 0 so that a(T(x), T(y)) < e, whenever p(x, y) < 5. Also T is continuous at every point in X if and only if, for every open set GCY, the set T~l(G) = {xeX : T(x) e G] is open.  Real Analysis - Seite 359von Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson - 1997 - 713 SeitenVollansicht - Über dieses Buch
 | Peng Yee Lee - 1989 - 194 Seiten
...condition holds for the Henstock integral. Lemma 3.4. A function f is Henstock integrable on [a,b] if and only if for every e > 0 there is a 5(£) > 0 such that for any S— fine divisions D - ([u,v];£) and D' - ([u'.v'];£'> we have |£ f(O(vu) - I... | |
 | Cyrus D. Cantrell - 2000 - 790 Seiten
...norm" of the matrix difference A' - A. In freshman calculus, one learns that a function / : R -> R is continuous at x if and only if for every e > 0, there exists a real number 8 > 0 such that | f(x') - f(x)\ < e for all x' such that x — x 8. In this example,... | |
 | Lee Peng Yee, Rudolf Vyborny - 2000 - 328 Seiten
...can offer. First we need LEMMA 2.6.1 (Alternative definition of derivative) F has a derivative f(x) at x if and only if for every e > 0 there is a 6 > 0 such that ifu^v and x-6<u<x<v<x+6 (2.55) then \F(v) - F(u) - f(x)(v - u)\ < e(v - u). (2.56)... | |
 | Brian S. Thomson, Andrew M. Bruckner, Judith B. Bruckner - 2008 - 408 Seiten
...else can you do? 2.7.9 In calculus courses one learns that a function / : M — > M is continuous at y if for every e > 0 there is a 5 > 0 so that \f(x) - f(y)\ < e for all \x — y\ < 5. Show that if / is continuous at y and sn — > y then f(sn)... | |
 | Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner - 2008 - 685 Seiten
...function with domain E and suppose that XQ is a point of accumulation of E. Then we write Hm /(*)=£, if for every e > 0 there is a 5 > 0 so that |/(*)- L|<e whenever x is a point of E differing from XQ and satisfying x — XQ < 5. Note. The condition... | |
 | Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner - 2001 - 753 Seiten
...domain E and suppose that XQ is a point of accumulation of E n (X0,0O). Then we write lim /(x) = L if for every e > 0 there is a <5 > 0 so that \f(x)-L\<e whenever XQ < x < XQ + 6 and x £ E. An equivalent sequential version can be established.... | |
 | Zdzisław Denkowski, Stanisław Migórski, Nikolaos Socrates Papageorgiou - 2003 - 708 Seiten
...convergent but not unconditionally convergent. Also from the above definition we have that £]¿e/ Xi = x if and only if for every e > 0, there is a finite set Je Ç I such that \\x — ^2i&JXi\\ < e for any finite set JC / such that JC\J£ = 0. Moreover,... | |
 | Peter Adams, P. Adams, Ken Smith, Rudolf V?born? - 2004 - 548 Seiten
...translates into the Cauchy definition of continuity as follows. Corollary 12.6.1 (to Theorem 12.6) / is continuous at x if and only if for every e > 0 there exists 6 > 0 such tnat \xx\<6=> \f(x) - f(x)\ < e. (12.9) It is a detail but an important one: the... | |
 | Steven George Krantz - 2005 - 324 Seiten
...and only if" form. For example A function / on an open interval / is said to be continuous at c £ I if and only if, for every e > 0, there is a <5 > 0 such that . . . for this convention is that we must teach our students to read definitions; the fact... | |
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