\hfill \hfill } %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.4in,right=0.5in,top=0.5in,bottom=0.4in} \pagestyle{fancy} \lhead{\color{blue} \large Math 207} \lfoot{\small \copyright $\;$ Hidegkuti, 2017} \cfoot{} \chead{\Large Continuous Functions} \rfoot{\footnotesize Last revised: September 19, 2017} \rhead{ \small page \thepage } \textwidth 7.56in \textheight 9.6in \setlength{\headheight}{29pt} \setlength{\parindent}{0pt} \setlength{\parskip}{0.08in} \begin{document} The following is the actual definition of a (finite) limit of a function $f$ at a number $c$.\vspace{0.08in} \textbf{Definition:} \ Suppose that $f$ is a function and $c$, $L$ are real numbers. \ We say that $\lim\limits_{x\rightarrow c}f\left( x\right) =L$ if for all $\varepsilon >0$ there exists $\delta >0$ such that for all $x\not=c$ with $\left\vert x-c\right\vert <\delta $, we also have that ($f$ is defined and) $\left\vert f\left( x\right) -L\right\vert <\varepsilon $.\vspace{0.08in% } In this course, we will not use the rigorous definition, we will use instead the following. \textbf{Definition:} \ If the left-hand side limit and the right-hand side limit both exist (and are finite) and are equal, we say that $% \lim\limits_{x\rightarrow c}f\left( x\right) =L$.\vspace{0.08in} Continuity is an extremely important property of functions that will have significant impact on other behaviors of functions. \ A function is continuous at a point if there is a two-sided, finite limit, and it is also the function value.\vspace{0.08in} \textbf{Definition:} \ A function $y=f\left( x\right) $ is \textbf{% continuous at a number }$c$ of its domain if the two-sided limit exists, $% f\left( c\right) $ exists, and $\lim\limits_{x\rightarrow c}f\left( x\right) =f\left( c\right) $.\vspace{0.08in} Continuity as deifned above is a local property, defined point by point. \ However, it will be beneficial to also define continuity on an interval.% \vspace{0.08in} \textbf{Definition:} \ (Continuity on an open interval). \ Suppose that $I$ is an open interval, i.e. $I=\left( a,b\right) $ or $I=\left( -\infty ,b\right) $ or $I=\left( a,\infty \right) $. \ A function $y=f\left( x\right) $ is continuous on $I$ if it is continuous at every real number $c$ that lies in $I.$\vspace{0.08in} We will also define continuity on a closed interval $\left[ a,b\right] $, as continuity on a closed interval will turn out to have extremely nice properties.\vspace{0.08in} \textbf{Definition: }\ (Continuity on a closed interval)\ \ A function $% y=f\left( x\right) $ is continuous on $\left[ a,b\right] $ if \qquad \qquad 1) \ $f$ is continuous at every number $c$ within the interval $\left( a,b\right) $. \ (A number $c$ is sometimes \newline %TCIMACRO{\TeXButton{white}{\color{white}}}% %BeginExpansion \color{white}% %EndExpansion .% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \qquad \qquad \qquad called an interior point of the interval). \qquad \qquad 2) \ $f$ is right-continuous at $x=a$, i.e. $% \lim\limits_{x\rightarrow a^{+}}f\left( x\right) $ exists, $f\left( a\right) $ exists, and $f\left( a\right) =\lim\limits_{x\rightarrow a^{+}}f\left( x\right) $ \qquad \qquad 3) \ $f$ is left-continuous at $x=b$, i.e. $% \lim\limits_{x\rightarrow b^{-}}f\left( x\right) $ exists, $f\left( b\right) $ exists, and $f\left( b\right) =\lim\limits_{x\rightarrow b^{-}}f\left( x\right) .$\bigskip Also note that another way to express continuity is to say that $% \lim\limits_{h\rightarrow 0}f\left( x+h\right) =f\left( x\right) $. \ Another alternative statement of continuity is $\lim\limits_{x\rightarrow c}f\left( x\right) =f\left( \lim\limits_{x\rightarrow c}x\right) $, so that there is a commutativity between taking the limit and taking the function values.\bigskip \textbf{Theorem:} \ Suppose that $f$ and $g$ are functions that are continuous at $x=c$. \ \ Then: \qquad 1) \ $f+g$ is continuous at $c$. \qquad 2) \ $fg$ is continuous at $c$ \qquad 3) \ $f-g$ is continuous at $c$ \qquad 4) \ If $g\left( c\right) \not=0,$ then $\dfrac{f}{g}$ is continuous at $c$ \qquad 5) If $g$ is continuos at $c$ and $f$ is continuous at $g\left( c\right) ,$ then $f\circ g$ is continuous at $c$.\bigskip \bigskip These properties can be proved using the properties of limits and the definitions of the functions $f+g$, $fg$, $f-g$, $\dfrac{f}{g}$ and $f\circ g $. \pagebreak \textbf{Example 1:} \ Suppose that $f$ is a function defined as $\ f\left( x\right) =\left\{ \begin{array}{ccc} mx-10 & \text{if} & x<-2 \\ x^{2}+9x-8 & \text{if} & x\geq -2% \end{array}% \right. .$ \ Find the value of $m$ if we know that $f$ is continuous everywhere.\vspace{0.07in} Solution: \ If $x<-2$, then the function is continuous for all $x.$ \ Similarly, $f$ is also continuous on all $x$ with $x\geq -2$. \ The only questionable point is at $x=-2.$ \ For a continuous function, we need the left limit and the right limit to exist and have the same value. \ \begin{eqnarray*} \lim_{x\rightarrow -2^{-}}f\left( x\right) &=&\lim_{x\rightarrow -2^{+}}f\left( x\right) \\ \lim_{x\rightarrow -2^{-}}\left( mx-10\right) &=&\lim_{x\rightarrow -2^{+}}\left( x^{2}+9x-8\right) \end{eqnarray*}% By the various properties of limits, this equation can be simplified as follows:% \begin{eqnarray*} m\left( -2\right) -10 &=&\left( -2\right) ^{2}+9\left( -2\right) -8 \\ -2m-10 &=&-22 \\ -2m &=&-12 \\ m &=&6 \end{eqnarray*}% And so $m=6$ is the value for which $f$ is continuous on the entire number line. \bigskip \bigskip \begin{center} {\Large Practice Problems}\bigskip \end{center} \begin{enumerate} \item Suppose that $f$ is a function defined as $\ f\left( x\right) =\left\{ \begin{array}{ccc} mx-13 & \text{if} & x<-10 \\ x^{2}+5x-3 & \text{if} & x\geq -10% \end{array}% \right. .$ \ Find the value of $m$ if we know that $f$ is continuous everywhere. \item Suppose that $f$ is a function defined as $\ f\left( x\right) =\left\{ \begin{array}{ccc} 8x-4 & \text{if} & x\leq 4 \\ -2x+b & \text{if} & x>4% \end{array}% \right. .$ \ Find the value of $b$ if we know that $f$ is continuous everywhere. \item Suppose that $f$ is a function defined as $\ f\left( x\right) =\left\{ \begin{array}{ccc} mx-11 & \text{if} & x<-6 \\ x^{2}+4x-5 & \text{if} & x\geq -6% \end{array}% \right. .$ \ Find the value of $m$ if we know that $f$ is continuous everywhere. \item Suppose that $f$ is a function defined as $\ f\left( x\right) =\left\{ \begin{array}{ccc} 2x+b & \text{if} & x<7 \\ \sqrt{x+2} & \text{if} & x\geq 7% \end{array}% \right. .$ \ Find the value of $b$ if we know that $f$ is continuous everywhere. \end{enumerate} \bigskip \bigskip \begin{center} {\Large Answers - Practice Problems\bigskip } \end{center} \bigskip 1.) \ $-6$ \ \ \ \ \ \ 2.) \ $36$\ \ \ \ \ \ \ 3.) \ $-3$ \ \ \ \ \ 4) \ $% -11 $\bigskip \bigskip {\small %TCIMACRO{\TeXButton{\small}{\small}}% %BeginExpansion \small% %EndExpansion } \href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{% For more documents like this, visit our page at\ https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}