证 .设[tex=1.857x1.357]bZ4KhrFbnCaidqbMGQZfww==[/tex]在[tex=2.0x1.357]dw9/rpY6rxhRs2Z3fy1fAg==[/tex]上处处可微,且导函数有限,则要证明对任意[tex=3.643x1.357]PpeCkscwUu0ZUnh67D9fYw==[/tex], 若[tex=2.643x1.0]l3Kv5dSgqCy7fQzYCh5JQw==[/tex]必有[tex=3.929x1.357]gCltqaruGVRta7jpKoi0PQ==[/tex]. 令 [tex=9.786x1.429]8P3kxH9I7zf3WBphDyHDVaKym1mQS2bUNjHGq1ucTO+YXlBXYgS9RIx3g/pRXby6f2o58bDOJy0gWksaspCxdg==[/tex]由于[tex=1.857x1.357]BGkv0wKMIn2R4tUsMDFEFA==[/tex]处处具有有限导数,所以[tex=4.071x3.286]5hElXtXzAL0GL5ie9+Cs/4GvvArcSmRNK9FgmZoNY9g=[/tex].从而[tex=6.5x3.286]VpPgjcKaZx93h9bQ3glpfQHyffoRSznt5blzmNWUhyQhLpYIa/BBONo2ynvnrs6u[/tex][tex=0.929x1.0]sU6vqnXtI3Bmxu0mmsYKxg==[/tex]又可分解为[tex=10.214x1.429]TiDBEAc7qh8fr8K6uTH2pPfI9h3jjyNSZ/zxGi3OtTlupdJxMxtOoi3zSqrpvPJs+FNxDJ94Rglz1WQcW+oRPQ==[/tex][tex=14.0x1.5]v40Tu2c7V7oDt9SUgOOiZmIQqWtKV9CNCL7lPuuqFB2ZF5SrSqbt3MOf37u+onwoykfpdmtQFkobdCoBtXoNOkwrk7U+cv5XdbZtC4fgM/yF/rORwnXkW6IauJr8SuBadXqyrPd3+DT4jhB41kAd8A==[/tex]于是[tex=10.5x3.286]VpPgjcKaZx93h9bQ3glpfR8zLR3Ad8BAX0bi9FC4uMn3Vbb8282NVZBvxGz77ZJlUTOxCgwd1DKvlkLyRej9muTIW9IyG014QpucrFDobyAcO2Z8P5cwWoaomqTbw91mw/Y1BN1xTqgIzK8+LjrbCA==[/tex]欲证[tex=4.429x1.286]PdjmRWGoVKm7wX/Qc3m5ng==[/tex], 只需证明对每个自然数[tex=0.643x1.286]ZsZs11iKEvfmzDIurZth8g==[/tex]有[tex=9.714x1.286]zLxt2/uUQI1LiRIu6fjjcTgwj6B9FBzieJYcn05OBHVgQMuAsw6vzBneKBZ1n/++2YDvK1ulIzPGW4GKyiseLBqDQckL9sstMkEg+82Kz50=[/tex]由于[tex=2.857x1.429]G4neOeFflDUvVgckD9Huurf2tZjzFhTRrgdIDj7BkWw=[/tex], 所以, 对任给定的[tex=2.357x1.071]NKdlHEPkxrjbs/M0XXsJWQ==[/tex], 必有开集[tex=4.5x1.429]AYwbgvqAuPLcw0Ax+pZgy0GW5i6n4FXH8C4NXEswMDwN++59vlhL5V5nweM45Nnx[/tex],且[tex=5.143x1.357]r8zRs8KFOBOF1wTdaV+lrid7TPL23deDiOCSpmFWbn2w3oizmanf737+gyZ1/1Uc[/tex].为简单起见记[tex=4.5x1.357]6P62Tn9QQVa/bdxXyBZNikNFt3KMRCmLWt7Fh05IuzQ=[/tex] .对任给的[tex=3.929x1.429]BP2YW3e0nWOutjum9DxT8Kfgmp2VGjGskJreT5U9ZjG8n05hOoDmixKQf+Z5XFf5[/tex], 由于[tex=6.0x1.429]WtrFWpm4hQQT3UgBjRwFPskoYFOrMrb1BOrdElof2kw=[/tex], 故存在[tex=2.786x1.214]r+tCZaiR3mIYYCELQeWGag==[/tex],当 [tex=5.286x1.357]7gKmaRB2yC55vAVNsAqxFA==[/tex]时,[tex=10.857x2.786]V1TqlEhFVISxrFWFeu5ZQXu//vMjySswbMPQesVj39Iam9iQLE7gKKcmYbtZk5vx[/tex]即[tex=12.643x1.357]okyPhEPeAJRzHBFeTqYzFGlvJKgGBpqqItu0DU4oURk=[/tex][tex=8.357x1.357]xFBMk3uCrq3iS8bgUxfcPA==[/tex](*)其中 [tex=0.929x1.214]UhCZNSL9+OH1WTDN31R77w==[/tex]可取得充分小, 使得[tex=8.571x1.357]PlVWN9imCOj1SwvjmROe5H5jYcfmVjxnAhebhrS+55CTE0uAroJ1xXi99RXDwAOB[/tex]对[tex=2.429x1.071]LIL5Hi9voB3EE4+rqDBHGQ==[/tex] ,[tex=2.857x1.214]wszbQ2QN2lU1dRHyHXZaCg==[/tex]作区间[tex=10.714x1.357]ItceDy7S9CuYG0nqkRYRI6pISkzhHCF/LPAJsBV3Bxg=[/tex],[tex=7.143x1.357]SUvTYYPr0zZqK8TsLUmRXQ==[/tex]及相应区间[tex=11.214x1.357]eLdvxoQl2BZMe8Bql4aLGA+2863Ao2i2e0RyXbzyq/lt9v8zpnBK3FNCkX5OrNgG[/tex] 由(*)知, [tex=11.643x1.357]2VuIc3Q910rekTQZKo6D8QicuOe+2Z0C0S7qUyDvB8k=[/tex],[tex=10.357x1.357]oIlAd/RstBiKOEJKhCr6S8oElvSJtw0NQ74E7IzzccE=[/tex] 显然[tex=13.786x1.857]7UVYy6tji2geVjuVhz64rdGQ4u4eHOLkAreVdqJffLWPR7BJjqfkbQzxEUnfN4WSUsgr/vE4r0PnopzNwqAa6W2EaCoWA7VG4AKzRZnCE4o=[/tex] 而且, 当[tex=10.857x1.357]vS2dUvnQ1PfEJEV96UTS8VMFDWVcL+73+kwMWkRj+GxG5sZaT7lGCxIqBPfjchjdtr7NuMHSXlLomYNE0Cl8T7/q8RxYE5kDlxwYepbl+Cg=[/tex]时,[tex=13.071x1.357]Ge2MaETeYxa+C3GZJGvQ4FNW/GRREPvTYNU961DCq89kkGyJSqdHU28j7l9Q6bIE5JWM4uw22yBVakzcBOXTsLVgjibJczGIETEODTdl/TfV2PgLoTO50eySXOBnp8Hr[/tex]从而[tex=10.214x1.357]DTg79XClOgEx8AhyGoqCySZxBjizeA10X3n/KpFCDKL8uVHfZnTHCw5D4Jcu21GqWjNcxBClDsEs6yLSDv6G4Q==[/tex]由上述结果可见区间组[tex=4.214x1.357]V0RMuIJGaoK4gd2zMyrRzNWjrF14Qs2woV6BykON8sU=[/tex]依维他利意义覆盖了[tex=2.143x1.429]rOFKVmakfHQMqvfXzjwLARXlmxK0MfFCrRD5zMajAns=[/tex]. 而[tex=2.143x1.429]rOFKVmakfHQMqvfXzjwLARXlmxK0MfFCrRD5zMajAns=[/tex]为有界集. (因为在[tex=0.714x1.429]oLpXDQMK+ZJLJeAlQGIBCQ==[/tex]上 [tex=2.214x1.429]i+dnt0m+Vi0IpEF4DSu/zA==[/tex]有界.)根据维他利定理,从[tex=4.214x1.357]V0RMuIJGaoK4gd2zMyrRzNWjrF14Qs2woV6BykON8sU=[/tex]可选到可列个两两不相交的区间[tex=4.857x1.357]YlhRpP3IzxyADhhT0g+suB/3bnRTXbNTPVugNA3TnM1CC4hPAAeKA6dAXmsCbeNO[/tex]使得[tex=13.143x3.357]gLqiKdu6UIiavcVLxPnlydSzJsLMlZbebbFmP2ecHZIkqp6Gg2YPpzPxEGsbJ54/7Edbleqs3YUrEokEEUfuMMSsSp3xSUmR5ivo5Q2jW0Ja2/jyZdUI1N7l9KcXAUZX+zSZ4jorpW1LREkSJGXy0A==[/tex]所以, [tex=17.929x16.0]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[/tex][br][/br]由[tex=4.857x1.357]YlhRpP3IzxyADhhT0g+suEa82nj0kaieFLaxT9cA2E1UfgNpulJtAE+y99Km1nFW[/tex]两两不相交,所以, [tex=4.571x1.357]jyrAvBVVWIA3lzBKDNBlMsHecQKXZRaqmhd/zCYpccEkZh4n17/HEMusPMU7p9X6[/tex]也是两两不相交的.从而 [tex=14.929x3.357]+BId2NK6oAAtGZHBvYVlz2/2XZcQ31kwcVu0/kMQn8Xs4+R17MH+rE7JXXvk6dNW3CKoKMBv+LJORtgC7z1PqE5/kO2Q1EF+YC0Qi4Z5PA+R3Rly9AFTaS7JNlFuwqD8NDIFeAOaPXHS8p1RLvTn76U+ebyl3W4prujqhDL20BU=[/tex]又由于 [tex=5.714x1.357]8E39mvjhJbI+EAldlXGVXOwpsB8Ju7v6AOm/zxZFdkw8LLEz4K0UvTR3kFkpVkCe[/tex], 对一切[tex=0.357x1.0]+eJLelx8thmbkEj/Y0iCOw==[/tex]成立.所以[tex=7.0x3.286]CoVqolDfpQtZYUa53dNnXwHUCmXwmYZQj6p8Yg2CdZmH2/I/1pDygaStyBfqllZCfhrDWWvGT7u9UpgnIKllKDcKBPt4WRNTwcOvK0tYpN0=[/tex],[tex=9.714x3.357]PamLm1gfJbfisRO/E1Ju3vDCBguR9+An72s9zs2jsvwtr9/2k6VaqVp9K5okke6wVfWZGqvQTMz53TTtkeSDze69hQADzmhXJGYBZ94OzOI=[/tex]从此可见 [tex=8.357x1.429]+BId2NK6oAAtGZHBvYVlz2/2XZcQ31kwcVu0/kMQn8XxUTCD9nTB0ITgy3Bbzwzkm45xDxfHwwtazoL17S3ppA==[/tex].又由[tex=0.5x0.786]OpoabfWfZdF4cYFv2GsywQ==[/tex]的任意性,立即得到[tex=4.857x1.429]+BId2NK6oAAtGZHBvYVlz2/2XZcQ31kwcVu0/kMQn8Wkn80ILKMI3jBbfjGSSo+W[/tex]同理可证 [tex=5.5x1.429]+BId2NK6oAAtGZHBvYVlz5zX5/oEF+ibeqezqqfX82jmxBGYazjrUDt9DUMSRD8Q[/tex].于是 [tex=4.286x1.357]hNbob09AIw5hHyt29ANAPQ==[/tex] .所以,对任意[tex=3.643x1.357]PpeCkscwUu0ZUnh67D9fYw==[/tex], 若[tex=2.643x1.0]l3Kv5dSgqCy7fQzYCh5JQw==[/tex], 有[tex=3.929x1.357]gCltqaruGVRta7jpKoi0PQ==[/tex]. 即[tex=1.857x1.357]BGkv0wKMIn2R4tUsMDFEFA==[/tex]具有性质[tex=1.643x1.357]dE6pM3s4g8wKfvnkaRESUg==[/tex].