设[tex=1.286x1.214]kPh+FHWBPmYJHd/Njak8uA==[/tex]均为定义在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上的有界函数 . 证 明: 若仅在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]中有限个点处[tex=5.0x1.286]6jKCdTr41kUom0l3mR9GJiNQLoO/Pk+UmN0thxrUN0M=[/tex],则当[tex=0.5x1.214]0K9Xf7VHWdVeOrSYAKIm6Q==[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上可积时[tex=0.5x1.0]wLRBXo571ziKptAIyBBTRQ==[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上也可积,且[tex=10.286x2.857]YQy8o6xXV2vuInKBm3FsSoz2Z90+AIx5XYsf6CImCA9WBnANclPis7+H2Nr/9GSQ[/tex]
证 不失一般性. 设[tex=1.286x1.214]kPh+FHWBPmYJHd/Njak8uA==[/tex]仅在一点[tex=0.5x0.786]EL0hSqs6jZBGdsmH7TMShQ==[/tex]处之值不同 , 这里[tex=3.071x1.357]RlpHfusYXFViH7nndfIOlw==[/tex]. 先证[tex=0.5x1.0]wLRBXo571ziKptAIyBBTRQ==[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上可积.由题设知[tex=1.286x1.214]kPh+FHWBPmYJHd/Njak8uA==[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上有界,故[tex=2.857x1.071]Encp8FQWq6GzGPXxxE/yqg==[/tex]使得[tex=4.357x1.357]aLWhKuOKpiuEWBgSM4vUwYAXW5Nd6YHgXXA1khBqPP8=[/tex]有[tex=4.714x1.357]r5VgX+CGoWyyhOQRhu4Ypw==[/tex],[tex=4.714x1.357]j2vUXoBtB/s2CECNuKajiA==[/tex]又[tex=0.5x1.214]0K9Xf7VHWdVeOrSYAKIm6Q==[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上可积,由可积的充要条件知 ,[tex=2.357x1.071]ghJXJtBd3IXQSpXoMpDSwV4Cy1fIU/jEONBwSa5nCVY=[/tex],[tex=1.286x1.286]0mtvsQVhCSKOyAhGsiZNbw==[/tex],使得[tex=6.357x2.714]fX5mNp8AKMzXyfblX/e4JAq+kiXfKgJBFHAWhxw8KAGvpgGHW99f2FlpyEIbfPAm9Bi9CQPlQYT9t1zvP69/1Nvl/NGCQUqappiSDTAMGBA=[/tex][br][/br]记[tex=1.0x1.286]gXRpKRR31sciC8QZX2YoSA==[/tex]为[tex=0.714x1.286]atrPPistVyxj7cY8rjePCQ==[/tex]的加密, 且满足[tex=6.286x2.214]vmlA360UN5+qprIAZfU+8gzUCBPCzrKCARXgGoYoOnT4+UiV8JhEu0RWZowm1HXwxWfgCvNDWEKyBVETJKIHUA==[/tex], 则知[br][/br][tex=11.857x2.714]uRgMU5ijU2iFCGJn88fWDgmwUuWQPlPsAOq34MoGVLHxJkFVoGsg7WFMfP4sDya/2ZR8Zj+WW6HfepAd1AlRF1GDtWMAGCsmq7GnNyrJwZ9Ox0jRp3N5WPn0v1RfEGaOMbFV+QdNWTffqRLG+11t3iI5FHRM9tb4+/fPtTHVwgJbpj6RyRxYZLo7u7DeOalD[/tex].[br][/br]设点[tex=0.5x0.786]EL0hSqs6jZBGdsmH7TMShQ==[/tex]落在[tex=1.0x1.286]gXRpKRR31sciC8QZX2YoSA==[/tex]中的第[tex=0.571x1.286]pc/qlnA3cxu8Ag9jp3tYHQ==[/tex]个小区间中,于是有[br][/br][tex=4.429x2.714]uRgMU5ijU2iFCGJn88fWDgmwUuWQPlPsAOq34MoGVLHOnUzTTlP9rtQyx6kqBF6OtI4a5fiyJfx0Y6WPOwoulw==[/tex][tex=10.429x2.5]eUe66FblmKUmiEx4jaTD1nBWB10eZe7J3/ONx0XolzvHiJvBjUwkFTdnUf181w7W++hcj88BjFcB7w9wu0vCsJYDGE+pelE3zvPy+pG2ly+pQDoFY0ofM735t1U5I0zRY50V50LNW7fFed4u5r+45Xg+XcKJxUUiKDkceBn0G+5UsPVFw/lGohFmvxoaOgRN[/tex][tex=5.643x2.143]a1WmT9b6NpFEVrrN7gzskajzhdhPih4Zmvk+EjRS4PsW9IdTxttiyQ5uvZ5R947YGyn2bfHEyWX7vcUkLGbc8gqjOiMzZUTc2AnORoUG1kg=[/tex][br][/br]而[tex=4.071x1.429]Zn28gOp/gC/PDPU4JbOdmEe+Nb+/MJwbNmm2sM7TNq4=[/tex],[tex=9.286x2.214]iPahI+iNFeGm5h6qBJWDfBp8kwAe9Uc/yWjtfXYkfCUo9KnQBQb/dHyWSy8pCviIFLfrZM6J9CShG2qUiNxIuMUrYYdsSfQFLkRYOKpBKXfCoJqwBGwOdagtvYYir8mg[/tex][br][/br]故 [tex=4.5x2.714]uRgMU5ijU2iFCGJn88fWDgmwUuWQPlPsAOq34MoGVLGDu+2K6mPancXgylfWsMmezvwLkYJ8VK3uSGQ787X4T1i9jrjNNiG2hfeLkBEQJ3c=[/tex][tex=7.643x2.214]a1WmT9b6NpFEVrrN7gzskdvAga1FsjNqXcediSxLiBMgSZ9kEPCyxRwngJRjGR3BPeZ1SxhUDzBxruuoO2k07g==[/tex][tex=5.071x2.143]a1WmT9b6NpFEVrrN7gzskQlbTBmNAXtTnOWpQA5WE7KAqx9JQoQO2dJEARL7C+zcnMC071hqp9tdNg0lOnIH/g==[/tex]所以可知 [tex=1.857x1.357]QPi3lZKJ+q/B5QY5cuDuQg==[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上可积.再证[tex=10.286x2.857]YQy8o6xXV2vuInKBm3FsSoz2Z90+AIx5XYsf6CImCA9WBnANclPis7+H2Nr/9GSQ[/tex][br][/br][br][/br]反证法 : 设[tex=6.286x2.857]YQy8o6xXV2vuInKBm3FsSrhXklQyqMVhXEXosUfTTKE=[/tex],[tex=6.214x2.857]51sEhS6hKIGQkehHgVQC0JfEX839xqMqT7geBt/t3cU=[/tex][br][/br][br][/br]且[tex=3.0x1.286]4giQG1VKc8EqoahjPJfEs0PFJ7SlGAt//9qnTkWDvVU=[/tex], [tex=4.714x1.357]r5VgX+CGoWyyhOQRhu4Ypw==[/tex], [tex=4.714x1.357]j2vUXoBtB/s2CECNuKajiA==[/tex],[tex=3.214x1.357]c+ORzTvnWYEdWlZXxFL1qQ==[/tex]，则由定积分定义知,[tex=5.5x2.429]wnnVw3AI26tAAle5k2adKx87gKNzP9n4rIOht0UBUiMo0HXUrlKBwZ/GKg9/Hd5AqnRaGBjYzy+4lmeqwgBwNw==[/tex],[tex=2.643x1.214]zlKeabRmX2Dxj8dDdgdiwCjG1cQR47j4RLY2/pAX76I=[/tex]使得对[tex=2.0x1.357]uQo0Qwms4Bgi6pleNWBbfw==[/tex]上的任何分割[tex=0.714x1.286]atrPPistVyxj7cY8rjePCQ==[/tex]及在其上任意选取的点集[tex=2.214x2.214]+/szbIIO8divZ+NfDkDTkt0l2/ZFzyKFKZpjMx2VUfk=[/tex], 只要[tex=3.714x1.357]bnLiOkavHYLhihLjEyyCG3XLbFIT4TgV8X/EeRGAN4A=[/tex], 就有[tex=10.786x3.357]JD6J+SxueeSIkrjnsgdmpIf5Rg/m/BjJ+wJbxyUfyDCDpGF7l4bc47Fr4e7GmOCk824wOTEzUa1MkvMxnAJbh/ldNvW/BBKSA0HZxE2sCep93X1LIBBZ02lBaIaTbrwOCbP7+6at6jJ0lPwdq2WzCA==[/tex].及[tex=3.214x1.214]jW96uFFyD7aVzVPbjH83opDWvxwCUumnaaNkdzj7O6Q=[/tex],使得对[tex=2.0x1.357]uQo0Qwms4Bgi6pleNWBbfw==[/tex]上的任何分割[tex=0.714x1.286]atrPPistVyxj7cY8rjePCQ==[/tex]及在其上任意选取的点集[tex=2.071x1.5]+/szbIIO8divZ+NfDkDTktpGuV7jbvHA/NJkJPs+XTs=[/tex]只要[tex=3.786x1.286]irCFTT4ECPJIsn/KAYMhfdAvlieHXxYsBsagL3UNTZc=[/tex]就有[tex=10.357x3.357]JD6J+SxueeSIkrjnsgdmpFY2lVKa9pPQ/6OdPidm4cpkHCfQn+bOFN5RfC2naL9AdXbgadeZ7jMmRMffABwTVaJnUqoXQ/OUKCsz9A7VK3RStjK7nrj8g6aUnwdvkhYm54zRx2/YbRMpZp6D80O0ag==[/tex].取[tex=10.357x2.786]GokL2oqu177VpjXg8Ks4Mtnc99HMJN4Zbjy8pyZy1Se426AtKRfMRndSzuWFlrxbFRKu/RHi0sRogYdMV4Ma2Q1Y3QjGl0y/Bck88Pnae6VJlux3k4cqEi7XEuljP0z9[/tex],则对于[tex=2.0x1.357]bXp5Vb63IyKXaWMS3BCP6w==[/tex]上的任何分割[tex=0.714x1.286]atrPPistVyxj7cY8rjePCQ==[/tex]及在其上任意选取的点集 [tex=1.714x1.357]DtWnREzVn/NKOc4OqzlknZcac97QRcoOFY/bntK7jxs=[/tex]只要[tex=3.429x1.357]Gl1TiAy9ggbeGYDGbfQi2w==[/tex]就有[tex=11.214x7.071]rZM5/OPAdr7aX+kNl9iwpNAukz97QVwBdApamxsbVGKJbuS3abI3ovjbOVuY80dFoUdAQzMiN6x86c9f0MI9/oTgO6W3sfclbQUMzwkh4YX0HQgcQaHoqGwLXVuW7iq9Mi5pobSClTO3Zgg2yGh1XlEEjlIpOWukR7LtP7tvJEUDA0wQt6bJFd4eH35/sMxrlW1wf9810qD4xtldP6J78NIxjA2a1CCVWoEh8tEM4qwKzSa3sqiBoaBcxO1rzbBZ4GAxc1ApwE7Sy8ReblE76DOnv6J/mBCLtPBX+r0e4XO/iaVWsWHYqt90LW6qJpH7[/tex][br][/br]同时成立，即[tex=18.857x3.357]JD6J+SxueeSIkrjnsgdmpIf5Rg/m/BjJ+wJbxyUfyDAsRNX08gPJxyYLqF6zJCUFanXbbn9QKXvqkvsRWAsXPOf5/0cihjiTxauAY42wlj4xlDzCuk9LDhqX5MKitBmS1V7afqPq+APoJZyg7K3JRni6n7BzpOjwhV92Z5+qgSHHoPEFvJN5QTGrl8AmtyHllzFtLx0plyrG7iCXZ0/AMQ==[/tex]而 [tex=17.643x13.929]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[/tex]由题设知[tex=1.857x1.357]bZ4KhrFbnCaidqbMGQZfww==[/tex]与[tex=1.857x1.357]fBOYuAIZ/H4m1Dx+my86tg==[/tex]仅在[tex=0.5x0.786]hycNLgozeED/VkKdun7zdA==[/tex]点不相等, 故[tex=9.143x3.286]AoNyWg9kh+1NTrJ36sPgAi0hGxI3rw+30PJhPqpTcTQarlJLGUVCvvtgjgahVByrYSrZmFjlowqtv/mRYZyFH4kGojFYKmt+CVGPmQevRYZDYFJQSYLJGivqUbGzDBNZ[/tex][tex=8.357x2.214]DNyg6jGL0xjDhU9isUipRR11nllYX7fH2OB9+JOVEnatJ4QKRkNLAI8wKQjUG/yAWRw/7Xsa5uu1+AC3pAQIfGJMqUXBA+k82Z5sI5W71Nw=[/tex]因此有[tex=7.286x2.143]N6rPSJoEPj5g+IaxMu3CFxwh3epcVkyZC/+z4RwTP4wTuQedb5El+haPDhXVm0B27UoM1/HSz1Qf/8N/JdjQ+of6isknRlYh8QEOUtp976o=[/tex],[br][/br]即[tex=10.786x2.357]N6rPSJoEPj5g+IaxMu3CF0y5jnGpT+rcbEelqBTFL/qAn0KgvIpIJyENaWYxEm2CrVDzRz4mrcnJ50a39dohyqaWji2mfucwV8mxczjMKObT47sPzWT1bSAMb/mGlpV9[/tex],[br][/br]矛盾, 从而[tex=2.357x1.214]2rL++P2+P7jAF2a4tKhiJg==[/tex]即[tex=10.571x2.857]NY7oodrirBbiImTnksGISeP5InpehyYXak28A033MDiZAnFjyHp+LGIj8OPUkrF8f1BrSP8UTjBYXOWvvxxaeg==[/tex]。[br][/br]综上,结论得证.
举一反三
- 设[tex=0.5x1.214]0K9Xf7VHWdVeOrSYAKIm6Q==[/tex]是[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上的可微函数， 试证明[tex=0.857x1.357]ZqaqzeJ1QgaSs6XNaCvPCg==[/tex]是[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上的可测函数。
- 证明: 若[tex=0.5x1.214]0K9Xf7VHWdVeOrSYAKIm6Q==[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上可积, [tex=0.643x1.0]J+LW/0i6Fe+lWEmBUgT8zg==[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上连续，且除有限个点外有[tex=5.0x1.429]vcoFvsi7S06HgXTsHmMDCSb7DcJ5ck9Hi0PzKAJUVzo=[/tex], 则有[br][/br][tex=10.071x2.857]YQy8o6xXV2vuInKBm3FsSlJYD2khGdpjb7CEWe0giR4=[/tex]
- 设[tex=2.214x1.429]i+dnt0m+Vi0IpEF4DSu/zA==[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上连续，在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex] 内可导, [tex=8.071x1.429]OAKNUgNilvva3jjhpGDuyHfXB6Vpb0HZ9tZUHbsSkn+T5T2iDUtIHpZ/3/r1gu9U[/tex], 证明: [tex=5.214x1.429]IjXlhkMhr9LwCKTDiko+hpKIZWd+1PbgIxo7JGm9Pr4=[/tex].
- 设[tex=1.857x1.357]BGkv0wKMIn2R4tUsMDFEFA==[/tex]是[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上的单调增加函数，且它的值域是整个区间[tex=4.357x1.357]+74dUdz21TTIpzOV3mq9pA==[/tex]，试证[tex=1.857x1.357]BGkv0wKMIn2R4tUsMDFEFA==[/tex]是[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上的连续函数。
- 设[tex=1.857x1.357]QPi3lZKJ+q/B5QY5cuDuQg==[/tex]是[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上的绝对连续函数，[tex=1.857x1.357]BGkv0wKMIn2R4tUsMDFEFA==[/tex]定义在[tex=4.643x1.357]3+NDETjbtRnj+mD3xG2zviOhqLdK3LTtKMvqcRw22dQ=[/tex]上，且满足利普希茨(Lipschitz)条件，试证明[tex=3.143x1.357]fUn7YZ664ewPCotsQ//Alg==[/tex]是[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上的绝对连续函数。
内容
- 0
设[tex=1.857x1.357]BGkv0wKMIn2R4tUsMDFEFA==[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上[tex=1.571x1.357]GtLNfoGLJZiK1mceQTIzpA==[/tex]可积，且处处有[tex=3.714x1.357]n3f7jwsT3zAd0hiq20ir9w==[/tex]，试证明[tex=8.286x2.857]LRrXvfh63hVL+k+pVfVbWjZWfERLtaNFxQKW5TU2MpojTALjR71TlHCh9Bj5HnQD[/tex]
- 1
设[tex=0.643x1.286]+RQz+inOZSqc5WvKyEpD0Q==[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上连续,且[tex=1.857x1.357]BGkv0wKMIn2R4tUsMDFEFA==[/tex]不恒等于零，证明[tex=7.5x2.857]kDMlGEM8OZNgVwa79ZgakASK+2WjHwMW2uUwJCxQYXw=[/tex]
- 2
设[tex=6.071x1.357]4LgfSyTm0h7bq9xqSv7G4LNyI9eXQ7t6nf458Xe37zjS4V81lBtGaGjd6xQTx6KK[/tex]在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上连续，在[tex=2.214x1.357]mpyYBdP7k8056w1o+qOOxw==[/tex]内可导，证明：存在[tex=3.286x1.357]EV4pc+LBkNBOhd4NZUA5NQ==[/tex]，使得[tex=16.357x3.357]Uyz5s0rmQIddjb5Jc2T/YRSnI70CPiP9kSoxG/LsBEQsOFwZaIYio/xDEuz4rvImZ3GEM+gn+IQRe1Rq9HOufnlmnLgQiRTLLWlyd2m5PpNiisTat4EvIByMSXzh8HVQPts8b6b0urtNgk9oqPzz4TXb9tDW2RcYR0g9UvBgcN8wfdw1empP7P1zbq+Jg68Psxp+i3UzfEuBVHPM7mUbhwNryRi3jXo3t34CuHeOXN8=[/tex]
- 3
证明：设[tex=3.714x1.357]1wcc6vqE76k/eJ2Xobhi2g==[/tex]，则[tex=0.5x1.214]0K9Xf7VHWdVeOrSYAKIm6Q==[/tex]能在[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上取得介于它的最大值[tex=1.0x1.0]/4LSvKfNeQWJ+IvWbbbjdA==[/tex]与最小值[tex=0.929x0.786]D9maNLyVVGrC3QbL9jjRWg==[/tex]之间的任一值。
- 4
证明[tex=1.857x1.357]BGkv0wKMIn2R4tUsMDFEFA==[/tex]为[tex=1.857x1.357]bawv/j+LZ1l+o4ciN/29dA==[/tex]上连续函数的充分必要条件是对任意实数[tex=0.5x0.786]EL0hSqs6jZBGdsmH7TMShQ==[/tex], 集[tex=6.143x1.357]CpKxGYq0bqZsw3HJXpFWZ1H5QJraiIjf+p3HFjeuvK4=[/tex]和[tex=6.714x1.357]CpKxGYq0bqZsw3HJXpFWZ9It7fIkG1Fhfd/gA4+TsOo=[/tex]常为闭集.