数学系毕业论文 - 图文(3)

fA (c)lim()(x)?,if B?0. x?pgB CONTINUOUS FUNCTIONS 4.5 Definition Suppose Xand Y are metric spaces, E?X,p?E, and f maps E into Y.Then f is said to be continuous at p if for every ??0there exists a ??0 such that dY(f(x),f(p))?? for all points x?E for which dX(x,p)??. If f is continuous at every point of E, then f is said to be constinuous on E. It shoule be noted that f has to be defined at the point p in order to be constinuous at p.(Compare this with the remark following Definition 4.1.) If p is an isolated point of E, then our definition implies that every function f which has E as its domain of definition is continuous at p. For, no matter which ??0we choose, we can pick??0so that the only point x?Efor which dX(x,p)??is x?p; then dY(f(x),f(p))?0??. 4.6 Theotem In the situation given in Definition 4.5, assume also that pis a limit point of E. Then f is continuous at p if and only if limf(x)?f(p). x?p4.7 Theorem Suppose X,Y,Z are metric spaces, E?X, f maps E into Y , g maps the range of f,f(E),into Z,and h is the mapping of E into Zdefined by h(x)?g(f(x)) (x?E) If f is continuous at a point p?E and if g is continuous at the point

f(p), then h is continuous at p. This function h is called the composition or the composite of f and g. The notation h?g?f is frequently used in this context. Proof Let ??0 be given. Since g is continuous at f(p), there exists ??0 such that dZ(g(y),g(f(p)))?? if dY(y,f(p))??and y?f(E) Since f is continuous at p, there exists ??0such that dY(f(x),f(p))?? if dX(x,p)??and x?E It follows that dZ(h(x),h(p))?dZ(g(f(x)),g(f(p)))?? if dX(x,p)?? and x?E.Thus h is continuous at p. DISCONTINUITIES If x is a point in the domain of definition of the function f at which f is not continuous, we say that f is discontinuous at x,or that f has a discontinuity at x.If fis defined on an interal or on a segment, it is customary to divide discontinuities into two types. Before giving this classification, we have to define the right-hand and the left-hand limits of f at x, which we denote by f(x?) and f(x?), respectively. 4.25 Definition Let f be difined on ?a,b?.Consider any point x such that a?x?b.We write f(x?)=q if f(tn)?q as n??, for all sequences ?tn? in ?x,b?such that tn?x. To obtain the definition of f(x?),for a?x?b,we restrict ourselves to sequences

?tn? in ?a,x?. It is clear that any point x of ?a,b?,limf(t) exists if and only if t?x f(x?)?f(x?)?limf(t). t?x4.26 Definition Let f be defined on ?a,b?.If f is discontinuous at a point x,and if f(x?) and f(x?) exist,then f is said to have a discontinuity of the first kind,or a simple discontinuity,at x.Otherwise the discontinuity is said to be of the second kind. There are two ways in which a function can have a simple discontinuity:either f(x?)?f(x?)?in which case the value f(x) is immaterial? Or f(x?)?f(x?)?f(x). 4.27 Examples (a) Define ?1 (x rational ), f(x)?? 0 (x irrational).? Then f has a discontinuity of the second kind at every point x,since neither f(x?) nor f(x?) exists. (b) Define ?x (x rational), f(x)?? 0 (x irrational).? Then f is continuous at x?0 and has a discontinuity of the second kind at every other point. (c) Define ?x?2 (?3?x??2),? f(x)???x?2 ( ?2?x?0 ), ?x?2 ( 0?x?1 ).? Then f has a simple discontinuity at x?0 and is continuous at every other point of ??3,1?. (d) Define

?1?sin (x?0), f(x)??x??0 (x?0). Since neither f(0?) nor f(0?) exists, f has a discontinuity of the second kind at x?0.We have not yet shown that sinx is a continuous function.If we assume this result for the moment,Theorem 4.7 implies that f is continuous at every point x?0.

本科生毕业论文设计

题目 函数的极限与连续

作者姓名 程雅 指导教师 谢永红 所在学院 数学与信息科学学院 专业（系） 数学与应用数学 班级（届） 07级c班

完成日期 2011 年 4 月 25 日

百度搜索“77cn”或“免费范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，免费范文网，提供经典小说综合文库数学系毕业论文 - 图文(3)在线全文阅读。