...for all li, m £ ,V. Simple examples (eg , the un(x) = x" of Problem 3, Section 1.5.3) show that the pointwise limit of a sequence of continuous functions need not be continuous. However, a sequence un(x) that satisfies the Cauchy criterion for uniform convergence will have a continuous...

...the questions we asked at the beginning of this chapter. We begin with continuity. We know that the pointwise limit of a sequence of continuous functions need not be continuous. We now show that the uniform limit of a sequence of continuous functions must be continuous. Theorem...

...the function f(x) defined on the unit interval by f(x) = 0 for 0 < x < 1 and /(I) = 1. Note that the pointwise limit of a sequence of continuous functions need not be continuous. 134 Example. We define a sequence of "tent functions" QJ on the unit interval by 9j(x) = J2z for 0...

...approach 0 as n approaches oo. In particular, lim fn(a) - lim a" n- >oo n->oo = 0 = /(a). Thus the pointwise limit of a sequence of continuous functions need not be continuous. 2.7 Uniform Convergence Definition 2.7.1 A sequence of functions /„ : [a,b] — > R will converge...