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 $f_1,f_2,f_3,...,f_k$ be real valued functions on a metric space X and $\underline{f}$ be a mapping from X on to $\mathbb{R}^k$ defined by $\underline{f}(x)=\left(f_1(x),f_2(x),f_3(x),...,f_k(x)\right)$, $x\in X$ then $\underline{f}$ is continuous on X if and only if $f_1,f_2,f_3,...,f_k$ 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