高等數(shù)學(xué)_同濟(jì)大學(xué)第六版--高等數(shù)學(xué)課件第一章函數(shù)與極限.pptVIP

,2019/6/29,函數(shù)與極限,2,一、基本概念,1.集合:,具有某種特定性質(zhì)的事物的總體.,組成這個(gè)集合的事物稱為該集合的元素.,有限集,無(wú)限集,2019/6/29,函數(shù)與極限,3,數(shù)集分類:,N-自然數(shù)集,Z-整數(shù)集,Q-有理數(shù)集,R-實(shí)數(shù)集,數(shù)集間的關(guān)系:,例如,不含任何元素的集合稱為空集.,例如,規(guī)定,空集為任何集合的子集.,2019/6/29,函數(shù)與極限,4,2.區(qū)間:,是指介于某兩個(gè)實(shí)數(shù)之間的全體實(shí)數(shù).這兩個(gè)實(shí)數(shù)叫做區(qū)間的端點(diǎn).,稱為開區(qū)間,稱為閉區(qū)間,2019/6/29,函數(shù)與極限,5,稱為半開區(qū)間,稱為半開區(qū)間,有限區(qū)間,無(wú)限區(qū)間,區(qū)間長(zhǎng)度的定義:,兩端點(diǎn)間的距離(線段的長(zhǎng)度)稱為區(qū)間的長(zhǎng)度.,2019/6/29,函數(shù)與極限,6,3.鄰域:,2019/6/29,函數(shù)與極限,7,4.常量與變量:,在某過(guò)程中數(shù)值保持不變的量稱為常量,注意,常量與變量是相對(duì)“過(guò)程”而言的.,通常用字母a, b, c等表示常量,而數(shù)值變化的量稱為變量.,常量與變量的表示方法：,用字母x, y, t等表示變量.,2019/6/29,函數(shù)與極限,8,5.絕對(duì)值:,運(yùn)算性質(zhì):,絕對(duì)值不等式:,2019/6/29,函數(shù)與極限,9,因變量,自變量,數(shù)集D叫做這個(gè)函數(shù)的定義域,二、函數(shù)概念,2019/6/29,函數(shù)與極限,10,自變量,因變量,對(duì)應(yīng)法則f,函數(shù)的兩要素:,定義域與對(duì)應(yīng)法則.,約定: 定義域是自變量所能取的使算式有意義的一切實(shí)數(shù)值.,2019/6/29,函數(shù)與極限,11,定義:,如果自變量在定義域內(nèi)任取一個(gè)數(shù)值時(shí)，對(duì)應(yīng)的函數(shù)值總是只有一個(gè)，這種函數(shù)叫做單值函數(shù)，否則叫與多值函數(shù),2019/6/29,函數(shù)與極限,12,(1) 符號(hào)函數(shù),幾個(gè)特殊的函數(shù)舉例,2019/6/29,函數(shù)與極限,13,(2) 取整函數(shù) y=x x表示不超過(guò) 的最大整數(shù),階梯曲線,2019/6/29,函數(shù)與極限,14,(3) 狄利克雷函數(shù),2019/6/29,函數(shù)與極限,15,(4) 取最值函數(shù),2019/6/29,函數(shù)與極限,16,2019/6/29,函數(shù)與極限,17,例1,脈沖發(fā)生器產(chǎn)生一個(gè)單三角脈沖,其波形如圖所示,寫出電壓U與時(shí)間 的函數(shù)關(guān)系式.,解,單三角脈沖信號(hào)的電壓,2019/6/29,函數(shù)與極限,18,2019/6/29,函數(shù)與極限,19,例2,解,故,2019/6/29,函數(shù)與極限,20,三、函數(shù)的特性,有界,無(wú)界,1函數(shù)的有界性:,2019/6/29,函數(shù)與極限,21,2函數(shù)的單調(diào)性:,2019/6/29,函數(shù)與極限,22,2019/6/29,函數(shù)與極限,23,3函數(shù)的奇偶性:,偶函數(shù),2019/6/29,函數(shù)與極限,24,奇函數(shù),2019/6/29,函數(shù)與極限,25,4函數(shù)的周期性:,（通常說(shuō)周期函數(shù)的周期是指其最小正周期）.,2019/6/29,函數(shù)與極限,26,直接函數(shù)與反函數(shù)的圖形關(guān)于直線 對(duì)稱.,四、反函數(shù),2019/6/29,函數(shù)與極限,27,五、小結(jié),基本概念 集合, 區(qū)間, 鄰域, 常量與變量, 絕對(duì)值.,函數(shù)的概念,函數(shù)的特性 有界性,單調(diào)性,奇偶性,周期性.,反函數(shù),2019/6/29,函數(shù)與極限,28,思考題,2019/6/29,函數(shù)與極限,29,思考題解答,設(shè),則,故,2019/6/29,函數(shù)與極限,30,練 習(xí) 題,2019/6/29,函數(shù)與極限,31,2019/6/29,函數(shù)與極限,32,練習(xí)題答案,2019/6/29,函數(shù)與極限,34,一、基本初等函數(shù),1.冪函數(shù),2019/6/29,函數(shù)與極限,35,2.指數(shù)函數(shù),2019/6/29,函數(shù)與極限,36,3.對(duì)數(shù)函數(shù),2019/6/29,函數(shù)與極限,37,4.三角函數(shù),正弦函數(shù),2019/6/29,函數(shù)與極限,38,余弦函數(shù),2019/6/29,函數(shù)與極限,39,正切函數(shù),2019/6/29,函數(shù)與極限,40,余切函數(shù),2019/6/29,函數(shù)與極限,41,正割函數(shù),2019/6/29,函數(shù)與極限,42,余割函數(shù),2019/6/29,函數(shù)與極限,43,5.反三角函數(shù),2019/6/29,函數(shù)與極限,44,2019/6/29,函數(shù)與極限,45,2019/6/29,函數(shù)與極限,46,冪函數(shù),指數(shù)函數(shù),對(duì)數(shù)函數(shù),三角函數(shù)和反三角函數(shù)統(tǒng)稱為基本初等函數(shù).,2019/6/29,函數(shù)與極限,47,二、復(fù)合函數(shù) 初等函數(shù),1.復(fù)合函數(shù),定義:,2019/6/29,函數(shù)與極限,48,注意:,1.不是任何兩個(gè)函數(shù)都可以復(fù)合成一個(gè)復(fù)合函數(shù)的;,2.復(fù)合函數(shù)可以由兩個(gè)以上的函數(shù)經(jīng)過(guò)復(fù)合構(gòu)成.,2.初等函數(shù),由常數(shù)和基本初等函數(shù)經(jīng)過(guò)有限次四則運(yùn)算和有限次的函數(shù)復(fù)合步驟所構(gòu)成并可用一個(gè)式子表示的函數(shù),稱為初等函數(shù).,2019/6/29,函數(shù)與極限,49,例1,解,2019/6/29,函數(shù)與極限,50,綜上所述,2019/6/29,函數(shù)與極限,51,三、雙曲函數(shù)與反雙曲函數(shù),奇函數(shù).,偶函數(shù).,1.雙曲函數(shù),2019/6/29,函數(shù)與極限,52,奇函數(shù),有界函數(shù),2019/6/29,函數(shù)與極限,53,雙曲函數(shù)常用公式,2019/6/29,函數(shù)與極限,54,2.反雙曲函數(shù),奇函數(shù),2019/6/29,函數(shù)與極限,55,2019/6/29,函數(shù)與極限,56,奇函數(shù),2019/6/29,函數(shù)與極限,57,四、小結(jié),函數(shù)的分類:,函數(shù),初等函數(shù),非初等函數(shù)(分段函數(shù),有無(wú)窮多項(xiàng)等函數(shù)),代數(shù)函數(shù),超越函數(shù),有理函數(shù),無(wú)理函數(shù),有理整函數(shù)(多項(xiàng)式函數(shù)),有理分函數(shù)(分式函數(shù)),2019/6/29,函數(shù)與極限,58,思考題,2019/6/29,函數(shù)與極限,59,思考題解答,不能,2019/6/29,函數(shù)與極限,60,一、填空題:,練 習(xí) 題,2019/6/29,函數(shù)與極限,61,2019/6/29,函數(shù)與極限,62,練習(xí)題答案,2019/6/29,函數(shù)與極限,63,2019/6/29,函數(shù)與極限,65,“割之彌細(xì)，所失彌少，割之又割，以至于不可割，則與圓周合體而無(wú)所失矣”,1、割圓術(shù)：,播放,劉徽,一、概念的引入,2019/6/29,函數(shù)與極限,66,正六邊形的面積,正十二邊形的面積,正 形的面積,2019/6/29,函數(shù)與極限,67,2、截丈問(wèn)題：,“一尺之棰，日截其半，萬(wàn)世不竭”,2019/6/29,函數(shù)與極限,68,二、數(shù)列的定義,例如,2019/6/29,函數(shù)與極限,69,注意：,1.數(shù)列對(duì)應(yīng)著數(shù)軸上一個(gè)點(diǎn)列.可看作一動(dòng)點(diǎn)在數(shù)軸上依次取,2.數(shù)列是整標(biāo)函數(shù),2019/6/29,函數(shù)與極限,70,播放,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,71,問(wèn)題:,當(dāng) 無(wú)限增大時(shí), 是否無(wú)限接近于某一確定的數(shù)值?如果是,如何確定?,問(wèn)題:,“無(wú)限接近”意味著什么?如何用數(shù)學(xué)語(yǔ)言刻劃它.,通過(guò)上面演示實(shí)驗(yàn)的觀察:,2019/6/29,函數(shù)與極限,72,2019/6/29,函數(shù)與極限,73,如果數(shù)列沒(méi)有極限,就說(shuō)數(shù)列是發(fā)散的.,注意：,2019/6/29,函數(shù)與極限,74,幾何解釋:,其中,2019/6/29,函數(shù)與極限,75,數(shù)列極限的定義未給出求極限的方法.,例1,證,所以,注意：,2019/6/29,函數(shù)與極限,76,例2,證,所以,說(shuō)明:常數(shù)列的極限等于同一常數(shù).,小結(jié):,用定義證數(shù)列極限存在時(shí),關(guān)鍵是任意給定 尋找N,但不必要求最小的N.,2019/6/29,函數(shù)與極限,77,例3,證,2019/6/29,函數(shù)與極限,78,例4,證,2019/6/29,函數(shù)與極限,79,四、數(shù)列極限的性質(zhì),1.有界性,例如,有界,無(wú)界,2019/6/29,函數(shù)與極限,80,定理1 收斂的數(shù)列必定有界.,證,由定義,注意：有界性是數(shù)列收斂的必要條件.,推論 無(wú)界數(shù)列必定發(fā)散.,2019/6/29,函數(shù)與極限,81,2.唯一性,定理2 每個(gè)收斂的數(shù)列只有一個(gè)極限.,證,由定義,故收斂數(shù)列極限唯一.,2019/6/29,函數(shù)與極限,82,例5,證,由定義,區(qū)間長(zhǎng)度為1.,不可能同時(shí)位于長(zhǎng)度為1的區(qū)間內(nèi).,2019/6/29,函數(shù)與極限,83,3.(收斂數(shù)列與其子數(shù)列間的關(guān)系) 如果數(shù)列 收斂于a，那么它的任一子數(shù)列也收斂，且極限也是a,2019/6/29,函數(shù)與極限,84,五.小結(jié),數(shù)列:研究其變化規(guī)律;,數(shù)列極限:極限思想,精確定義,幾何意義;,收斂數(shù)列的性質(zhì):有界性唯一性.,2019/6/29,函數(shù)與極限,85,思考題,證明,要使,只要使,從而由,得,取,當(dāng) 時(shí)，必有 成立,2019/6/29,函數(shù)與極限,86,思考題解答,（等價(jià)）,證明中所采用的,實(shí)際上就是不等式,即證明中沒(méi)有采用“適當(dāng)放大” 的值,2019/6/29,函數(shù)與極限,87,從而 時(shí)，,僅有 成立，,但不是 的充分條件,反而縮小為,2019/6/29,函數(shù)與極限,88,練 習(xí) 題,2019/6/29,函數(shù)與極限,89,“割之彌細(xì)，所失彌少，割之又割，以至于不可割，則與圓周合體而無(wú)所失矣”,1、割圓術(shù)：,劉徽,一、概念的引入,2019/6/29,函數(shù)與極限,90,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,91,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,92,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,93,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,94,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,95,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,96,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,97,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,98,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,99,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,100,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,101,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,102,三、數(shù)列的極限,2019/6/29,函數(shù)與極限,104,播放,一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限,2019/6/29,函數(shù)與極限,105,通過(guò)上面演示實(shí)驗(yàn)的觀察:,問(wèn)題:,如何用數(shù)學(xué)語(yǔ)言刻劃函數(shù)“無(wú)限接近”.,2019/6/29,函數(shù)與極限,106,2019/6/29,函數(shù)與極限,107,2.另兩種情形:,2019/6/29,函數(shù)與極限,108,3.幾何解釋:,2019/6/29,函數(shù)與極限,109,例1,證,2019/6/29,函數(shù)與極限,110,二、自變量趨向有限值時(shí)函數(shù)的極限,2019/6/29,函數(shù)與極限,111,2019/6/29,函數(shù)與極限,112,2.幾何解釋:,注意：,2019/6/29,函數(shù)與極限,113,例2,證,例3,證,2019/6/29,函數(shù)與極限,114,例4,證,函數(shù)在點(diǎn)x=1處沒(méi)有定義.,2019/6/29,函數(shù)與極限,115,例5,證,2019/6/29,函數(shù)與極限,116,3.單側(cè)極限:,例如,2019/6/29,函數(shù)與極限,117,左極限,右極限,2019/6/29,函數(shù)與極限,118,左右極限存在但不相等,例6,證,2019/6/29,函數(shù)與極限,119,三、函數(shù)極限的性質(zhì),1.有界性,2.唯一性,2019/6/29,函數(shù)與極限,120,推論,3.不等式性質(zhì),定理(保序性),2019/6/29,函數(shù)與極限,121,定理(保號(hào)性),推論,2019/6/29,函數(shù)與極限,122,4.子列收斂性(函數(shù)極限與數(shù)列極限的關(guān)系),定義,定理,2019/6/29,函數(shù)與極限,123,證,2019/6/29,函數(shù)與極限,124,例如,函數(shù)極限與數(shù)列極限的關(guān)系,函數(shù)極限存在的充要條件是它的任何子列的極限都存在,且相等.,2019/6/29,函數(shù)與極限,125,例7,證,2019/6/29,函數(shù)與極限,126,二者不相等,2019/6/29,函數(shù)與極限,127,四、小結(jié),函數(shù)極限的統(tǒng)一定義,(見下表),2019/6/29,函數(shù)與極限,128,2019/6/29,函數(shù)與極限,129,思考題,2019/6/29,函數(shù)與極限,130,思考題解答,左極限存在,右極限存在,不存在.,2019/6/29,函數(shù)與極限,131,一、填空題:,練 習(xí) 題,2019/6/29,函數(shù)與極限,132,2019/6/29,函數(shù)與極限,133,練習(xí)題答案,2019/6/29,函數(shù)與極限,134,一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限,2019/6/29,函數(shù)與極限,135,一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限,2019/6/29,函數(shù)與極限,136,一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限,2019/6/29,函數(shù)與極限,137,一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限,2019/6/29,函數(shù)與極限,138,一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限,2019/6/29,函數(shù)與極限,139,一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限,2019/6/29,函數(shù)與極限,140,一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限,2019/6/29,函數(shù)與極限,141,一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限,2019/6/29,函數(shù)與極限,142,一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限,2019/6/29,函數(shù)與極限,144,一、無(wú)窮小,1.定義:,極限為零的變量稱為無(wú)窮小.,2019/6/29,函數(shù)與極限,145,例如,注意,1.無(wú)窮小是變量,不能與很小的數(shù)混淆;,2.零是可以作為無(wú)窮小的唯一的數(shù).,2019/6/29,函數(shù)與極限,146,2.無(wú)窮小與函數(shù)極限的關(guān)系:,證,必要性,充分性,2019/6/29,函數(shù)與極限,147,意義,1.將一般極限問(wèn)題轉(zhuǎn)化為特殊極限問(wèn)題(無(wú)窮小);,3.無(wú)窮小的運(yùn)算性質(zhì):,定理2 在同一過(guò)程中,有限個(gè)無(wú)窮小的代數(shù)和仍是無(wú)窮小.,證,2019/6/29,函數(shù)與極限,148,注意 無(wú)窮多個(gè)無(wú)窮小的代數(shù)和未必是無(wú)窮小.,2019/6/29,函數(shù)與極限,149,定理3 有界函數(shù)與無(wú)窮小的乘積是無(wú)窮小.,證,2019/6/29,函數(shù)與極限,150,推論1 在同一過(guò)程中,有極限的變量與無(wú)窮小的乘積是無(wú)窮小.,推論2 常數(shù)與無(wú)窮小的乘積是無(wú)窮小.,推論3 有限個(gè)無(wú)窮小的乘積也是無(wú)窮小.,都是無(wú)窮小,2019/6/29,函數(shù)與極限,151,二、無(wú)窮大,絕對(duì)值無(wú)限增大的變量稱為無(wú)窮大.,2019/6/29,函數(shù)與極限,152,特殊情形：正無(wú)窮大，負(fù)無(wú)窮大,注意,1.無(wú)窮大是變量,不能與很大的數(shù)混淆;,3. 無(wú)窮大是一種特殊的無(wú)界變量,但是無(wú)界變量未必是無(wú)窮大.,2019/6/29,函數(shù)與極限,153,不是無(wú)窮大,無(wú)界，,2019/6/29,函數(shù)與極限,154,證,2019/6/29,函數(shù)與極限,155,三、無(wú)窮小與無(wú)窮大的關(guān)系,定理4 在同一過(guò)程中,無(wú)窮大的倒數(shù)為無(wú)窮小;恒不為零的無(wú)窮小的倒數(shù)為無(wú)窮大.,證,2019/6/29,函數(shù)與極限,156,意義 關(guān)于無(wú)窮大的討論,都可歸結(jié)為關(guān)于無(wú)窮小的討論.,2019/6/29,函數(shù)與極限,157,四、小結(jié),1、主要內(nèi)容:,兩個(gè)定義;四個(gè)定理;三個(gè)推論.,2、幾點(diǎn)注意:,無(wú)窮小與無(wú)窮大是相對(duì)于過(guò)程而言的.,（1） 無(wú)窮?。?大）是變量,不能與很小（大）的數(shù)混淆，零是唯一的無(wú)窮小的數(shù)；,（2）無(wú)窮多個(gè)無(wú)窮小的代數(shù)和（乘積）未必是無(wú)窮小.,（3） 無(wú)界變量未必是無(wú)窮大.,2019/6/29,函數(shù)與極限,158,思考題,2019/6/29,函數(shù)與極限,159,思考題解答,不能保證.,例,有,2019/6/29,函數(shù)與極限,160,一、填空題:,練 習(xí) 題,2019/6/29,函數(shù)與極限,161,2019/6/29,函數(shù)與極限,162,練習(xí)題答案,2019/6/29,函數(shù)與極限,164,一、極限運(yùn)算法則,定理,證,由無(wú)窮小運(yùn)算法則,得,2019/6/29,函數(shù)與極限,165,2019/6/29,函數(shù)與極限,166,推論1,常數(shù)因子可以提到極限記號(hào)外面.,推論2,有界，,2019/6/29,函數(shù)與極限,167,二、求極限方法舉例,例1,解,2019/6/29,函數(shù)與極限,168,小結(jié):,2019/6/29,函數(shù)與極限,169,解,商的法則不能用,由無(wú)窮小與無(wú)窮大的關(guān)系,得,例2,2019/6/29,函數(shù)與極限,170,解,例3,(消去零因子法),2019/6/29,函數(shù)與極限,171,例4,解,(無(wú)窮小因子分出法),2019/6/29,函數(shù)與極限,172,小結(jié):,無(wú)窮小分出法:以分母中自變量的最高次冪除分子,分母,以分出無(wú)窮小,然后再求極限.,2019/6/29,函數(shù)與極限,173,例5,解,先變形再求極限.,2019/6/29,函數(shù)與極限,174,例6,解,2019/6/29,函數(shù)與極限,175,例7,解,左右極限存在且相等,2019/6/29,函數(shù)與極限,176,三、小結(jié),1.極限的四則運(yùn)算法則及其推論;,2.極限求法;,a.多項(xiàng)式與分式函數(shù)代入法求極限; b.消去零因子法求極限; c.無(wú)窮小因子分出法求極限; d.利用無(wú)窮小運(yùn)算性質(zhì)求極限; e.利用左右極限求分段函數(shù)極限.,2019/6/29,函數(shù)與極限,177,思考題,在某個(gè)過(guò)程中，若 有極限， 無(wú)極限，那么 是否有極限？為什么？,2019/6/29,函數(shù)與極限,178,思考題解答,沒(méi)有極限,假設(shè) 有極限，,有極限，,由極限運(yùn)算法則可知：,必有極限，,與已知矛盾，,故假設(shè)錯(cuò)誤,2019/6/29,函數(shù)與極限,179,一、填空題:,練 習(xí) 題,2019/6/29,函數(shù)與極限,180,二、求下列各極限:,2019/6/29,函數(shù)與極限,181,2019/6/29,函數(shù)與極限,182,練習(xí)題答案,2019/6/29,函數(shù)與極限,184,一、無(wú)窮小的比較,例如,極限不同, 反映了趨向于零的“快慢”程度不同.,不可比.,觀察各極限,2019/6/29,函數(shù)與極限,185,定義:,2019/6/29,函數(shù)與極限,186,例1,解,例2,解,2019/6/29,函數(shù)與極限,187,常用等價(jià)無(wú)窮小:,用等價(jià)無(wú)窮小可給出函數(shù)的近似表達(dá)式:,例如,2019/6/29,函數(shù)與極限,188,二、等價(jià)無(wú)窮小替換,定理(等價(jià)無(wú)窮小替換定理),證,2019/6/29,函數(shù)與極限,189,例3,解,不能濫用等價(jià)無(wú)窮小代換.,對(duì)于代數(shù)和中各無(wú)窮小不能分別替換.,注意,2019/6/29,函數(shù)與極限,190,例4,解,解,錯(cuò),2019/6/29,函數(shù)與極限,191,例5,解,2019/6/29,函數(shù)與極限,192,三、小結(jié),1.無(wú)窮小的比較:,反映了同一過(guò)程中, 兩無(wú)窮小趨于零的速度快慢, 但并不是所有的無(wú)窮小都可進(jìn)行比較.,2.等價(jià)無(wú)窮小的替換:,求極限的又一種方法, 注意適用條件.,高(低)階無(wú)窮小; 等價(jià)無(wú)窮小; 無(wú)窮小的階.,2019/6/29,函數(shù)與極限,193,思考題,任何兩個(gè)無(wú)窮小量都可以比較嗎？,2019/6/29,函數(shù)與極限,194,思考題解答,不能,例當(dāng) 時(shí),都是無(wú)窮小量,但,不存在且不為無(wú)窮大,故當(dāng) 時(shí),2019/6/29,函數(shù)與極限,195,練 習(xí) 題,2019/6/29,函數(shù)與極限,196,2019/6/29,函數(shù)與極限,197,2019/6/29,函數(shù)與極限,198,練習(xí)題答案,2019/6/29,函數(shù)與極限,199,2019/6/29,函數(shù)與極限,201,一、函數(shù)的連續(xù)性,1.函數(shù)的增量,2019/6/29,函數(shù)與極限,202,2.連續(xù)的定義,2019/6/29,函數(shù)與極限,203,2019/6/29,函數(shù)與極限,204,例1,證,由定義2知,2019/6/29,函數(shù)與極限,205,3.單側(cè)連續(xù),定理,2019/6/29,函數(shù)與極限,206,例2,解,右連續(xù)但不左連續(xù) ,2019/6/29,函數(shù)與極限,207,4.連續(xù)函數(shù)與連續(xù)區(qū)間,在區(qū)間上每一點(diǎn)都連續(xù)的函數(shù),叫做在該區(qū)間上的連續(xù)函數(shù),或者說(shuō)函數(shù)在該區(qū)間上連續(xù).,連續(xù)函數(shù)的圖形是一條連續(xù)而不間斷的曲線.,例如,2019/6/29,函數(shù)與極限,208,例3,證,2019/6/29,函數(shù)與極限,209,二、函數(shù)的間斷點(diǎn),2019/6/29,函數(shù)與極限,210,1.跳躍間斷點(diǎn),例4,解,2019/6/29,函數(shù)與極限,211,2.可去間斷點(diǎn),例5,2019/6/29,函數(shù)與極限,212,解,注意 可去間斷點(diǎn)只要改變或者補(bǔ)充間斷處函數(shù)的定義, 則可使其變?yōu)檫B續(xù)點(diǎn).,2019/6/29,函數(shù)與極限,213,如例5中,跳躍間斷點(diǎn)與可去間斷點(diǎn)統(tǒng)稱為第一類間斷點(diǎn).,特點(diǎn),2019/6/29,函數(shù)與極限,214,3.第二類間斷點(diǎn),例6,解,2019/6/29,函數(shù)與極限,215,例7,解,注意 不要以為函數(shù)的間斷點(diǎn)只是個(gè)別的幾個(gè)點(diǎn).,2019/6/29,函數(shù)與極限,216,狄利克雷函數(shù),在定義域R內(nèi)每一點(diǎn)處都間斷,且都是第二類間斷點(diǎn).,僅在x=0處連續(xù), 其余各點(diǎn)處處間斷.,2019/6/29,函數(shù)與極限,217,在定義域 R內(nèi)每一點(diǎn)處都間斷, 但其絕對(duì)值處處連續(xù).,判斷下列間斷點(diǎn)類型:,2019/6/29,函數(shù)與極限,218,例8,解,2019/6/29,函數(shù)與極限,219,三、小結(jié),1.函數(shù)在一點(diǎn)連續(xù)必須滿足的三個(gè)條件;,3.間斷點(diǎn)的分類與判別;,2.區(qū)間上的連續(xù)函數(shù);,第一類間斷點(diǎn):可去型,跳躍型.,第二類間斷點(diǎn):無(wú)窮型,振蕩型.,間斷點(diǎn),(見下圖),2019/6/29,函數(shù)與極限,220,可去型,第一類間斷點(diǎn),跳躍型,無(wú)窮型,振蕩型,第二類間斷點(diǎn),2019/6/29,函數(shù)與極限,221,思考題,2019/6/29,函數(shù)與極限,222,思考題解答,且,2019/6/29,函數(shù)與極限,223,但反之不成立.,例,但,2019/6/29,函數(shù)與極限,224,練 習(xí) 題,2019/6/29,函數(shù)與極限,225,2019/6/29,函數(shù)與極限,226,練習(xí)題答案,2019/6/29,函數(shù)與極限,227,2019/6/29,函數(shù)與極限,229,一、四則運(yùn)算的連續(xù)性,定理1,例如,2019/6/29,函數(shù)與極限,230,二、反函數(shù)與復(fù)合函數(shù)的連續(xù)性,定理2 嚴(yán)格單調(diào)的連續(xù)函數(shù)必有嚴(yán)格單調(diào)的連續(xù)反函數(shù).,例如,反三角函數(shù)在其定義域內(nèi)皆連續(xù).,2019/6/29,函數(shù)與極限,231,定理3,證,2019/6/29,函數(shù)與極限,232,將上兩步合起來(lái):,2019/6/29,函數(shù)與極限,233,意義,1.極限符號(hào)可以與函數(shù)符號(hào)互換;,例1,解,2019/6/29,函數(shù)與極限,234,例2,解,同理可得,2019/6/29,函數(shù)與極限,235,定理4,注意 定理4是定理3的特殊情況.,例如,2019/6/29,函數(shù)與極限,236,三、初等函數(shù)的連續(xù)性,三角函數(shù)及反三角函數(shù)在它們的定義域內(nèi)是連續(xù)的.,2019/6/29,函數(shù)與極限,237,定理5 基本初等函數(shù)在定義域內(nèi)是連續(xù)的.,(均在其定義域內(nèi)連續(xù) ),定理6 一切初等函數(shù)