Section 1.3 Continuity
Recall that a function \(f\) is continuous at a number \(a\) if \(\displaystyle \lim_{x\to a}f(x)=f(a)\text{.}\) Alternatively, a function \(f\) is continuous at a number \(a\) if
The function \(f\) is defined at the number \(a\text{;}\)
The limit \(\lim_{x\to a}f(x)\) exists;
\(\lim_{x\to a}f(x)=f(a)\text{.}\)
Solve the following problems:
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Given the function
\begin{equation*} f(x)=\left\{ \begin{array}{lll} c-x\amp \mbox{if} \amp x\leq \pi \\ c\sin x\amp \mbox{if} \amp x> \pi \end{array} \right. \end{equation*}Find the value of the constant \(c\) so that the function \(f(x)\) is continuous.
For the value of \(c\) found above verify whether the 3 conditions for continuity are satisfied.
Draw a graph of \(f(x)\) from \(x=-\pi\) to \(x=3\pi\) indicating the scaling used.
Use the Intermediate Value Theorem to show that \(\displaystyle 2^x=\frac{10}{x}\) for some \(x>0\text{.}\)
Show that the equation \(\displaystyle 2^x=\frac{10}{x}\) has no solution for \(x\lt 0\text{.}\)
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Sketch a graph of the function
\begin{equation*} f(x)=\left\{ \begin{array}{lll} 2-x^2\amp \mbox{if} \amp 0\leq x\lt 1 \\ \frac{5}{2}\amp \mbox{if} \amp x=1\\ |2-x|\amp \mbox{if} \amp 1\lt x\leq 3 \\ \frac{1}{x-3}\amp \mbox{if} \amp 3\lt x\leq 5 \\ 2+\sin (2\pi x)\amp \mbox{if} \amp 5\lt x\leq 6 \\ 2\amp \mbox{if} \amp x> 6 \end{array} \right. \end{equation*}Answer the following questions by TRUE or FALSE:
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Is \(f\) continuous at:
\(x=1\text{?}\)
\(x=6\text{?}\)
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Do the following limits exist?
\(\displaystyle \displaystyle \lim _{x\to 1}f(x)\)
\(\displaystyle \displaystyle \lim _{x\to 3^-}f(x)\)
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Is \(f\) differentiable
at \(x=1\text{?}\)
on \((1,3)\text{?}\)
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Assume that
\begin{equation*} f(x)=\left\{ \begin{array}{lll} 2+\sqrt{x}\amp \mbox{if} \amp x\geq 1 \\ \frac{x}{2}+\frac{5}{2}\amp \mbox{if} \amp x\lt 1 \end{array} \right. \end{equation*}Determine whether or not \(f\) is continuous at \(x=1\text{.}\) Justify your answer and state your conclusion.
Using the definition of the derivative, determine \(f'(1)\text{.}\)
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Find the value of \(b\) so that the function
\begin{equation*} f(x)=\left\{ \begin{array}{ll} x^3+bx-7\amp \mbox{if } x\leq 2\\ be^{x-2}\amp \mbox{if } x>2 \end{array} \right. \end{equation*}is continuous everywhere. Justify your answer.
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Find the value of \(a\in \mathbb{R}\) so that the function
\begin{equation*} f(x)=\left\{ \begin{array}{ll} ax^2+9\amp \mbox{if } x> 0\\ x^3+a^3\amp \mbox{if } x\geq 0 \end{array} \right. \end{equation*}is continuous.
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Find the value of \(a\in \mathbb{R}\) so that the function
\begin{equation*} f(x)=\left\{ \begin{array}{ll} ax^2+2x\amp \mbox{if } x\lt 2\\ x^3-ax\amp \mbox{if } x\geq 2 \end{array} \right. \end{equation*}is continuous.
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Determine \(a\in \mathbb{R}\) such that the function
\begin{equation*} f(x)\left\{ \begin{array}{ll} \frac{\cos x}{2x-\pi}\amp \mbox{if } x> \frac{\pi}{2}\\ ax\amp \mbox{if } x\leq \frac{\pi}{2} \end{array} \right. \end{equation*}is continuous everywhere. Justify your answer.
Give one example of a function \(f(x)\) that is continuous for all values of \(x\) except \(x=3\text{,}\) where it has a removable discontinuity. Explain how you know that \(f\) is discontinuous at \(x=3\text{,}\) and how you know that the discontinuity is removable.
Sketch the graph of a function that has a removable discontinuity at \(x=2\) and an infinite discontinuity at \(x=7\text{,}\) but is continuous everywhere else.
Sketch the graph of a function that has a jump discontinuity at \(x=3\) and a removable discontinuity at \(x=5\text{,}\) but is continuous everywhere else.
A function \(h:I\to I\) is said to have a fixed point at \(x=c\in I\) if \(h(c)=c\text{.}\) Suppose that the domain and range of a function \(f(x)\) are both the interval \([0,1]\) and that \(f\) is continuous on its domain with \(f(0)\not=0\) and \(f(1)\not=1\text{.}\) Prove that \(f\) has at least one fixed point, i.e. prove that \(f(c)=c\) for some \(c\in(0,1)\text{.}\)