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# Chapter 03 - Limits and Continuity

- Limit of the function, examples and definition
- Theorem: Suppose (i) $(X,{d_x})$ and $(Y,{d_y})$ be two metric spaces (ii) $E\subset X$ (iii) $f:E\to Y$ i.e.
*f*maps*E*into*X*(iv)*p*is the limit point of*E*. Then $\lim_{x\to p} f(x)=q$ iff $\lim_{n\to\infty}f(p_n)=q$ for every sequence {$p_n$} in*E*such that ${p_n}\ne p$, $\lim_{n\to\infty}{p_n}=p$. - Examples and exercies
- Theorem: If $\lim_{x\to c}f(x)$ exists then it is unique.
- Theorem: Suppose that a real valued function
*f*is defined on an open interval*G*except possibly at $c\in G$. Then $\lim_{x\to c}f(x)=l$ if and only if for every positive real number $\varepsilon$, there is $\delta>0$ such that $|f(t)-f(s)|<\varepsilon$ whenever*s*and*t*are in $\left\{x:|x-c|<\delta \right\}$. - Theorem (Sandwiching Theorem): Suppose that
*f*,*g*and*h*are functions defined on an open interval*G*except possibly at $c\in G$. Let $f\le h\le g$ on*G*. If $\lim_{x\to c}f(x)=\lim_{x\to c}g(x)=l$, then $\lim_{x\to c}h(x)=l$. - Theorem: (for sum, difference, product of limit of fuctions)
- Continuity (in term of metric spaces)
- Theorem: Let (i)
*X, Y, Z*be metric spaces (ii) $E\subset X$ (iii) $f:E\to Y$, $g:f(E)\to Z$ and $h:E\to Z$ defined by $h(x)=g\left(f(x)\right)$. If*f*is continuous at $p\in E$ and if*g*is continuous at the point $f(p)$, then*h*is continuous at*p*. - Theorem: Let
*f*be defined on*X*. If*f*is continuous at $c\in X$ then there exists a number $\delta>0$ such that*f*is bounded on the open interval $(c-\delta ,c+\delta)$. - Theorem: Suppose
*f*is continuous 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 $f(x)>0$ for all $x\in G$. - Theorem: A mapping of a metric space
*X*into a metric space*Y*is continuous on*X*iff $f^{-1}(V)$ is open in*X*for every open set*V*in*Y*. - Theorem: Let $f_1,f_2,f_3,\ldots,f_k$ be real valued functions on a metric space
*X*and $\underline{f}$ be a mapping from*X*on to $\mathbb{R}^k$ defined by $\underline{f}(x)=\left(f_1(x),f_2(x),f_3(x),\ldots,f_k(x)\right)$, $x\in X$ then $\underline{f}$ is continuous on*X*if and only if $f_1,f_2,f_3,\ldots,f_k$ are continuous on*X*. - Theorem: Suppose
*f*is continuous on [*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) If $f(a)>0$ and $f(b)<0$ then there is a point*c*, $a<c<b$ such that $f(c)=0$. - Theorem (The intermediate value theorem): Suppose
*f*is continuous on [*a*,*b*] and $f(a)\neq f(b)$, then given a number $\lambda$ that lies between $f(a)$ and $f(b)$, there exists a point $c, a<c<b$ with $f(c)=\lambda$. - Theorem: Suppose
*f*is continuous on [*a*,*b*], then*f*is bounded on [*a*,*b*] (Continuity implies boundedness). - Uniform continuity and examples
- Theorem: If
*f*is continuous on a closed and bounded interval [*a*,*b*], then*f*is uniformly continuous on [*a*,*b*]. - Theorem: Let $\underline{f}$ and $\underline{g}$ be two continuous mappings from a metric space
*X*into $\mathbb{R}^k$, then the mappings $\underline{f}+\underline{g}$ and $\underline{f} \cdot \underline{g}$ are also continuous on*X*, i.e. the sum and product of two continuous vector valued function are also continuous. - Discontinuities

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