实变函数论

（重定向自实分析

內容

連續函數

${\displaystyle f\colon I\rightarrow \mathbf {R} .}$

${\displaystyle I=(a,b)=\{x\in \mathbf {R} \,|\,a

${\displaystyle I=[a,b]=\{x\in \mathbf {R} \,|\,a\leq x\leq b\}.}$

級數

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2^{n}}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots =1}$.

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots =\infty }$.

（此處“${\displaystyle =\infty }$”不是嚴謹的表示方式，只是表示部份和會無限制地増長）

微分

${\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}$

積分

黎曼積分

${\displaystyle a=x_{0}\leq t_{1}\leq x_{1}\leq t_{2}\leq x_{2}\leq \cdots \leq x_{n-1}\leq t_{n}\leq x_{n}=b.\,\!}$

${\displaystyle \sum _{i=1}^{n}f(t_{i})\Delta _{i};}$

${\displaystyle \left|S-\sum _{i=1}^{n}f(t_{i})\Delta _{i}\right|<\varepsilon .}$

参考资料

1. ^ Gaughan, Edward. 1.1 Sequences and Convergence. Introduction to Analysis. AMS (2009). ISBN 0-8218-4787-2.
2. ^ Stewart, James. Calculus: Early Transcendentals 6th. Brooks/Cole. 2008. ISBN 0-495-01166-5.