黎曼函数极限证明思路

2023-04-11 13:02 作者:仕方無 0人读过 | 我要投稿

Riemann function，黎曼函数（变体）

$%5Cbegin%7Bequation%7D%0AR(x)%3D%5Cleft%5C%7B%0A%5Cbegin%7Baligned%7D%0A%260%20%5Cquad%20x%E4%B8%BA%E6%97%A0%E7%90%86%E6%95%B0%20%5C%5C%0A%26%5Cfrac%7B1%7D%7Bp%7D%20%5Cquad%20x%3D%5Cfrac%7Bq%7D%7Bp%7D%20%2Cp%5Cin%20N%5E%2B%2Cq%5Cin%20Z%2C%20p%E4%B8%8Eq%E4%BA%92%E8%B4%A8%20%5C%5C%0A%261%20%5Cquad%20x%3D0%20%5C%5C%0A%5Cend%7Baligned%7D%20%0A%5Cright%0A.%0A%5Cend%7Bequation%7D$

$%E8%AF%81%E6%98%8E%EF%BC%9A%5Cforall%20x_%7B0%7D%5Cin%20%5B0%2C1%5D%2C%5Clim_%7Bx%5Cto%20x_%7B0%7D%20%7D%20R(x)%3D0%E3%80%82$

证明思路：

$%E5%85%88%E5%86%99%E5%87%BA%5Cvarepsilon%20-%20%5Cdelta%20%E8%AF%AD%E8%A8%80%EF%BC%9A$

$%5Clim_%7Bx%5Cto%20x_%7B0%7D%20%7D%20R(x)%3D0%E6%BB%A1%E8%B6%B3%EF%BC%9A$

$%5Ccolor%7Bred%7D%7B%5Cforall%20%5Cvarepsilon%20%3E0%7D%EF%BC%8C%5Cexists%20%5Cdelta%20%3E0%EF%BC%8C%5Cforall%20x(0%3C%5Cvert%20x-x_%7B0%7D%20%5Cvert%20%3C%5Cdelta%20)%EF%BC%8C%5Cvert%20R(x)-0%20%5Cvert%3C%5Cvarepsilon%20%20%EF%BC%8C$

$%5Cbecause%20R(x)%5Cgeq%200%EF%BC%8C%5Ctherefore%20%E5%8F%AF%E5%8C%96%E4%B8%BAR(x)%3C%5Cvarepsilon%20%EF%BC%8C%E5%8D%B3R(x)%3D%5Cfrac%7B1%7D%7Bp%7D%3C%5Cvarepsilon%20%20%3D%5Cfrac%7B1%7D%7Bk%7D%20(%E4%BB%A4%5Cvarepsilon%20%3D%5Cfrac%7B1%7D%7Bk%7D%20)%EF%BC%8C$

$%5Cdelta%20%E6%AD%A4%E6%97%B6%E4%B8%80%E5%AE%9A%E5%AD%98%E5%9C%A8%E4%BD%86%E5%B9%B6%E4%B8%8D%E5%A5%BD%E6%89%BE%E4%BB%A5%E7%94%A8%E4%BA%8E%E8%AF%81%E6%98%8E%E3%80%82$

$%E5%8F%8D%E8%BF%87%E6%9D%A5%E6%83%B3%EF%BC%8C%E5%BD%93R(x)%5Cgeq%20%5Cvarepsilon%20%E6%97%B6%EF%BC%8C%E6%98%AF%E6%89%BE%E4%B8%8D%E5%88%B0%E4%B8%80%E4%B8%AA%E7%AC%A6%E5%90%880%3C%5Cvert%20x-x_%7B0%7D%20%5Cvert%20%3C%5Cdelta%20%E7%9A%84%5Cdelta%E7%9A%84%E3%80%82$

$%E7%8E%B0%E4%BB%A4%E5%A4%A7%E4%BA%8E%E7%AD%89%E4%BA%8E%5Cvarepsilon%20%E7%9A%84%E6%9C%80%E5%B0%8F%E7%9A%84R(x)%E7%9A%84%E5%80%BC%E4%B8%BA%5Cfrac%7B1%7D%7Ba%7D(a%5Cin%20N%5E%2B%20)%20%EF%BC%8C%E6%9C%89%5Cfrac%7B1%7D%7Ba%7D%5Cleq%20%5Cfrac%7B1%7D%7Bk%7D%3C%20%20%5Cfrac%7B1%7D%7Ba%2B1%7D%20%20%E3%80%82$

$%E5%88%99%E6%AD%A4%E6%97%B6%5Cfrac%7B1%7D%7Ba%2B1%7D%20%E6%98%AF%E5%B0%8F%E4%BA%8E%5Cvarepsilon%20%E7%9A%84%E6%9C%80%E5%A4%A7R(x)%E5%80%BC%E3%80%82$

$%E7%8E%B0%E5%9C%A8%E5%8F%AF%E4%BB%A5%E6%89%BE%E5%88%B0%E4%B8%80%E4%B8%AA%5Cdelta%20%E4%BD%BF%E5%BE%97%5Cforall%20x(0%3C%5Cvert%20x-x_%7B0%7D%20%5Cvert%20%3C%5Cdelta)%EF%BC%8C%E6%9C%89R(x)%3C%5Cvarepsilon%E3%80%82$

$%E5%BD%93R(x)%3D%5Cfrac%7B1%7D%7Ba%2B1%7D%E6%97%B6%EF%BC%8Cx%E6%9C%80%E5%A4%9A%E5%8F%AF%E5%8F%96a%2B1%E4%B8%AA%E5%80%BC%EF%BC%8C%E5%8D%B3x%E5%8F%AA%E8%83%BD%E5%8F%96%E6%9C%89%E9%99%90%E4%B8%AA%E5%80%BC%E3%80%82$

$%E5%9C%A8%E8%BF%99a%2B1%E4%B8%AA%E6%9C%89%E9%99%90%E7%9A%84%E5%80%BC%E4%B8%AD%EF%BC%8C%E5%BF%85%E8%83%BD%5Ccolor%7Bred%7D%7B%E6%89%BE%E5%88%B0%E4%B8%80%E4%B8%AA0%3C%5Cmin_%7B1%5Cleq%20i%20%5Cleq%20a%2B1%7D%5Cleft%5C%7B%20%5Cvert%20x_%7Bi%7D-x_%7B0%7D%20%5Cvert%20%20%5Cright%5C%7D%20%3C%5Cdelta%7D%EF%BC%8C$ $%5Ccolor%7Bred%7D%7B%E4%BD%BF%E5%BE%97%7D%5Cfrac%7B1%7D%7Ba%2B1%7D%20%3D%5Ccolor%7Bred%7D%7BR(x)%3C%5Cvarepsilon%7D%20%3D%5Cfrac%7B1%7D%7Bk%7D%20%E3%80%82$

标红处就是满足极限条件的部分。

网上关于黎曼函数的证明有很规范的步骤，我这里只是作为一个解题者给出了正向思考的过程。

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