就说f在p点连续. 如果f在E的每一点都连续，就说f在E上连续。 应该注意，要使f在点p连续，f必须在点p有定义.(这一点请与定义4.1后面的说明对比一下) 如果p是E的一个孤立点，那么由我们的定义推知，每一个以E为定义域的函数都在点p连续。因为不管取的哪个??0，总可以选一个??0，使得满足dX(x,p)??的点x?E只有x=p;于是dY(f(x),f(p))?0??. 4.6 定理 在定义4.5里所假定的情况下，再假定p是E的极限点。那么，f在点p连续当且仅当 limf(x)?f(p) x?p 4.7定理 设X,Y,Z是度量空间，E?X，f将E映入Y内，g将f的值域f(E)映入Z内，而h是由 h(x)?g(f(x)) （x?E） 定义的E到Z内的映射。如果f在点p?E连续，并且g在点f(p)连续，那么h在点p连续。 这个函数h叫做f与g的复合函数或f和g合成。记号 h?g?f 在本书中经常用。 证明： 设??0已经给定。因为g在f(p)连续，便有??0使得当dY(y,f(p))??和y?f(E)有dZ(g(y),g(f(p)))??。又因为f在p点连续，那么存在着??0，使得当dX(x,p)??和x?E有dY(f(x),f(p))??。由此知道：当所以h在点p连dX(x,p)??和x?E有dZ(h(x),h(p))?dZ(g(f(x)),g(f(p)))??。续。 间断 如果x是函数f的定义域中的一点，而在这点f不连续，那么我们说f在x间断。如果f定义在一个闭区间或定义在一个开区间上，那么习惯上把间断分为两类。在讲这分类之前，我们必须定义f在x的右极限和左极限，对此，分别用
f(x?)和f(x?)表示它们。 4.25 定义 设f定义在?a,b?上，考虑任一点x，a?x?b。如果对于(x,b)中一切满足tn?x的序列?tn?来说，f(tn)?q（当n??），那么我们就写成 f(x?)?q. 为了对于a?x?b，得到f(x?)的定义，我们就把序列?tn?限制在?a,x?之内。 显然，在?a,b?的任一点x，limf(t)存在，当且仅当 t?x f(x?)?f(x?)?limf(t). t?x4.26 定义 设f定义在?a,b?上，如果f在一点间断，并且如果f(x?)和f(x?)都存在，就说f在x发生了第一类间断，或简单间断。其他的间断称为第二类间断。 函数发生简单间断的方式有两种：f(x?)?f(x?)，（在这种情况下数值f(x)无关紧要）或f(x?)?f(x?)?f(x)。 4.27 例 （a）定义 ?1 （当x是有理数）, f(x)?? ?0 （当x是无理数）.这时f在每个点x发生一次第二类间断，因为f(x?)和f(x?)都不存在。 （b）定义 ?x (当x是有理数) f(x)?? ?0 (当x是无理数).这时f在x?0连续，而在每个其他的点发生第二类间断。 （c）定义 ? x?2 (?3?x??2),? f(x)???x?2 (?2?x?0), ?x?2 (0?x?1).?这时f在x?0发生一次简单间断，而在??3,1?的其他每个点连续。 (d)定义
?1?sin (x?0), f(x)??x ??0 (x?0).因为f(0?)和f(0?)都不存在，所以f在x?0发生一次第二类间断。我们尚未证明sinx是连续函数。如果我们暂时承认这个结果，那么定理4.7就说明f在每个点x?0连续。
英文原版 LIMITS OF FUNCTIONS 4.1 Definition Let X and Y be metric spaces;suppose E?X,f maps E into Y,and p is a limit of E.We write f(x)?q as x?p,or (1) limf(x)?q x?p if there is a point q?Y with the following property: For every ??0 there exists a ??0 such that (2) dY(f(x),q)?? For all points x?E for which (3) 0?dX(x,p)??. The symbols dX and dY refer to the distances in Xand Y,respectively. If X and/or Yare replaced by the real line,the complex plane, or by some euclidean space Rk,the distance dX,dY are of course replaced by absolute values, or by norms of differences. It should be noted that p?X, but that p need not be a point of E in the above definition. Moreover, even if p?E,we may very well have f(p)?limf(x). x?p We can recast this definition in terms of limits of sequences: 4.2 Theorem Let X,Y,E,fand pbe as in Definition 4.1. Then (4) limf(x)?q x?pIf and only if (5) limf(pn)?q n??For every sequence ?pn?in E such that (6) pn?p，limpn?p. n?? Proof Suppose (4) holds. Choose ?pn?in E satisfying (6). Let ??0be given.Then there exists ??0such that dY(f(x),q)?? if x?Eand 0?dX(x,p)??. Also, there exists N such that n?N implies
0?dX(pn,p)??. Thus, for n?N,we have dY(f(pn),q)??, which shows that (5) holds. Conversely, suppose (4) is false. Then there exists some ??0 such that for every ??0there exists a pointx?E(depending on ?),for which 1（n=1,2,3…）, we thus dY(f(x),q)?? but 0?dX(x,p)??.Taking ?n?，nfind a sequence in E satisfying (6) for which (5) is false. Corollary Iffhas a limit atp, this limit is unique. 4.3 Definition Suppose we have two complex functions, fandg,both defined on E.Byf?gwe mean the function which assigns to each point x of E the numberf(x)?g(x).Similarly we define the difference f?g,the product fg,and the quotient f/gof the two functions,with the understanding that the quotient is defined only at those points x of E at which g(x)?0. If f assigns to each point x of E the same number c, then f is said to be a constant function, or simply a constant, and we write f?c. If fand gare real functions, and if f(x)?g(x) for every x?E, we shall sometimes write f?g, for brevity. Similarly, if f and g map E into Rk, we define f?g and f?g by (f?g)(x)?f(x)?g(x), (f?g)(x)?f(x)?g(x); and if ? is a real number, (?f)(x)??f(x). 4.4 Theorem Suppose E?X, a metric space, pis a limit point of E, f and g are complex functions on E, and limf(x)?A,limg(x)?B. x?px?p Then (a) lim(f?g)(x)?A?B; x?p (b) lim(fg)(x)?AB; x?p