楊明順

(渭南師范學(xué)院數(shù)學(xué)與信息科學(xué)學(xué)院,陜西渭南 714099)

關(guān)于四類函數(shù)的幾個(gè)結(jié)論

楊明順

(渭南師范學(xué)院數(shù)學(xué)與信息科學(xué)學(xué)院,陜西渭南 714099)

利用初等及組合方法對(duì)連續(xù)函數(shù)、單調(diào)函數(shù)、有界變差函數(shù)、絕對(duì)連續(xù)函數(shù)的關(guān)系進(jìn)行了探討,得出了絕對(duì)連續(xù)函數(shù)一定是有界變差函數(shù),單調(diào)函數(shù)是有界變差函數(shù),絕對(duì)連續(xù)函數(shù)一定是連續(xù)函數(shù)的結(jié)論.

連續(xù)函數(shù);單調(diào)函數(shù);有界變差函數(shù);絕對(duì)連續(xù)函數(shù)

連續(xù)函數(shù)、單調(diào)函數(shù)、有界變差函數(shù)、絕對(duì)連續(xù)函數(shù)是函數(shù)論中廣泛使用的四類函數(shù),搞清楚它們之間的關(guān)系對(duì)于掌握這些函數(shù)很有必要.目前有關(guān)研究它們之間關(guān)系的完整結(jié)果很少看到,本文就此問題進(jìn)行了探究.

定義1[1]設(shè)f(x)是定義在點(diǎn)集E?R上的函數(shù),x0∈E,若對(duì)?ε＞0,?δ＞0,當(dāng)x∈E∩U(x0, δ)時(shí),有f(x)-f(x0)＜ε,則稱f(x)在x0上連續(xù),如果f(x)在E上每點(diǎn)都連續(xù),就稱f(x)在E上連續(xù).

定義2[1]設(shè)f(x)是定義在實(shí)直線R中點(diǎn)集E上的有限函數(shù),如果對(duì)任意x1,x2∈E,當(dāng)x1＜x2時(shí),不等式f(x1)≤f(x2)恒成立,就稱f(x)是E上的單調(diào)增加函數(shù),如果不等式f(x1)＜f(x2)恒成立,就稱f(x)是E上的嚴(yán)格單調(diào)增加函數(shù);如果當(dāng)x1＜x2時(shí),不等式f(x1)≥f(x2)恒成立,就稱f(x)是E上的單調(diào)減少函數(shù),如果不等式f(x1)＞f(x2)恒成立,就稱f(x)是E上的嚴(yán)格單調(diào)減少函數(shù).

1.絕對(duì)連續(xù)函數(shù)一定是有界變差函數(shù)

2.單調(diào)函數(shù)是有界變差函數(shù)

3.絕對(duì)連續(xù)函數(shù)一定是連續(xù)函數(shù)

4.連續(xù)函數(shù)不一定是有界變差函數(shù)

5.有界變差函數(shù)不一定是連續(xù)函數(shù)

6.有界變差函數(shù)不一定是絕對(duì)連續(xù)函數(shù)

7.有界變差函數(shù)不一定是單調(diào)函數(shù)

例4 設(shè)f(x)=x,g(x)=x2,在[0,1]上都單調(diào)增加,令:H(x)=f(x)-g(x),x∈[0,1],則H(x)在[0,1]上有界變差函數(shù),但H(x)在[0,1]上不是單調(diào)函數(shù).

8.絕對(duì)連續(xù)函數(shù)不一定是單調(diào)函數(shù)

絕對(duì)連續(xù)函數(shù)一定是有界變差函數(shù),而有界變差函數(shù)不一定是單調(diào)函數(shù),從而可知:絕對(duì)連續(xù)函數(shù)不一定是單調(diào)函數(shù).

9.單調(diào)函數(shù)不一定是絕對(duì)連續(xù)函數(shù)

10.單調(diào)函數(shù)不一定是連續(xù)函數(shù)

11.連續(xù)函數(shù)不一定是單調(diào)函數(shù)

例7 設(shè)f(x)=x2,則f(x)在定義域(-∞,+∞)內(nèi)是連續(xù)的,但不是單調(diào)函數(shù).

12.連續(xù)函數(shù)不一定是絕對(duì)連續(xù)函數(shù)

[1]趙靜輝,徐吉華.實(shí)變函數(shù)簡(jiǎn)明教程(函數(shù)論部分)[M].武漢:華中科技大學(xué)出版社,1996.86-96.

[2]胡適耕.實(shí)變函數(shù)(微分論部分)[M].北京:高等教育出版社,1997.153-160.

[3]張曉嵐.實(shí)變函數(shù)與泛函分析簡(jiǎn)明教程[M].北京:高等教育出版社,2000.65-75.

[4]張喜堂.實(shí)變函數(shù)論的典型問題與方法[M].武漢:華中師范大學(xué)出版社,2000.307-320.

[5]胡長(zhǎng)松.實(shí)變函數(shù)(有界變差函數(shù))[M].北京:科學(xué)出版社,2002.133-134.

【責(zé)任編輯 牛懷崗】

On Several Conclusions about Four Kinds of Functions

YANG Ming-shun
(School of Mathematics and Information Science,Weinan Normal University,Weinan 714099,China)

The paper,based on the elementary function and the combinatorialmethod,discussed the relation between the continuous function,themonotonic function,the bounded variation function and the definitely continuous function.It proved that the absolute continuous functionmust be the bounded variation function,monotone function is a function of bounded variation,and absolutely continuous function is a continuous function.

continuous function;monotonic function;bounded variation function;definitely continuous function

O156.4

A

1009-5128(2014)03-0016-03

2013-09-30

陜西省教育廳科學(xué)研究計(jì)劃(自然科學(xué)專項(xiàng))項(xiàng)目:關(guān)于F.Smarandache數(shù)論問題的計(jì)算機(jī)輔助研究(2013JK1165);渭南師范學(xué)院教改項(xiàng)目:大學(xué)數(shù)學(xué)教學(xué)改革的探索與實(shí)踐(JG201345)

楊明順(1964—),男,陜西渭南人,渭南師范學(xué)院數(shù)學(xué)與信息科學(xué)學(xué)院教授,主要從事數(shù)論研究.