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- Chapter 04 - Differentiation
- * Theorem (Chain Rule) * Examples * Local Maximum * Theorem: Let //f// be defined on [//a//,//b//], if //f// has a local maximum at a point $x\in [a,b]$ and if $f'(x)$ exist then $f'(x)=0$. (The analogous for local minimum is o... }$ be differentiable in (//a//,//b//) then there exists $x\in (a,b)$ such that $\left|\underline{f}(b)
- Chapter 03 - Limits and Continuity
- and exercies * Theorem: If $\lim_{x\to c}f(x)$ exists then it is unique. * Theorem: Suppose that a... /. If //f// is continuous at $c\in X$ then there exists a number $\delta>0$ such that //f// is bounded... b]. If $f(c)>0$ for some $c\in [a,b]$ then there exist an open interval $G \subset[a,b]$ such that $f(... mbda$ that lies between $f(a)$ and $f(b)$, there exists a point $c, a<c<b$ with $f(c)=\lambda$. * Th
- Chapter 01 - Real Number System
- t lower bound property. * Field. * Proofs of axioms of real numbers. * Ordered field. * Theorems on ordered field. * Existence of real field. * Theorem: (a) Archimedean... property (b) Between any two real numbers there exits a rational number. * Theorem: Given two real
- Chapter 02 - Sequence and Series
- the sequence $\{s_n\}$ converges to //s// then $\exists$ a positive integer such that $\left| {\,{s_n}... Theorem: For each irrational number //x//, there exists a sequence $\left\{{r_n}\right\}$ of distinct ... y if for any real number $\varepsilon >0$, there exists a positive integer $n_0$ such that $\left|\sum