利用有理数对实数逼近的表示方式，给出黎曼函数处处不可导的一种证明，给出单位圆周上的有理点在单位圆上稠密的证明。

Rational number can approximate to real number, use the notation of approximate one can prove Riemann function isn t differentiable anywhere, that the Rational points are dense in unit circle.

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摘要利用有理数对实数逼近的表示方式，给出黎曼函数处处不可导的一种证明，给出单位圆周上的有理点在单位圆上稠密的证明。

Rational number can approximate to real number use the notation of approximate one can prove riemann function isn't differentiable anywhere that the rational points are dense in unit circle .

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在构造了完备化空间之后，证明了该空间就是勒贝格可积函数空间，从而说明了黎曼积分的完备化形式是勒贝格积分。

After constrcting the perfective space prove that this space is just the space of lebes gue integratiable function thus explain that lebes gue integral is the form of the perfective riemann integral.

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