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Chapter 03 - Limits and Continuity
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if for every positive real number $\varepsilon$, there is $\delta>0$ such that $|f(t)-f(s)|<\varepsilon$... on //X//. If //f// is continuous at $c\in X$ then there exists a number $\delta>0$ such that //f// is bou... on [a, b]. If $f(c)>0$ for some $c\in [a,b]$ then there exist an open interval $G \subset[a,b]$ such that... n [//a//,//b//] (i) If $f(a)<0$ and $f(b)>0$ then there is a point //c//, $a<c<b$ such that $f(c)=0$. (ii
Chapter 01 - Real Number System
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====== ==== Contents & Summary ==== * Theorem: There is no rational //p// such that $p^2=2$. * Theor... imedean property (b) Between any two real numbers there exits a rational number. * Theorem: Given two real numbers //x// and //y//, $x<y$ there is an irrational number //u// such that $x<u<y$. * Theorem: For every real number //x// there is a set //E// of rational number such that $x=\s
Chapter 02 - Sequence and Series
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$. * Theorem: For each irrational number //x//, there exists a sequence $\left\{{r_n}\right\}$ of disti... and only if for any real number $\varepsilon >0$, there exists a positive integer $n_0$ such that $\left|
Chapter 04 - Differentiation
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rline{f}$ be differentiable in (//a//,//b//) then there exists $x\in (a,b)$ such that $\left|\underline{f