1
Give an example of a function (you can write a formula, or sketch a graph) that has infinitely many infinite discontinuities.
Subsection 1.6.4 Exercises
Exercises — Stage 1
2
When I was born, I was less than one meter tall. Now, I am more than one meter tall. What is the conclusion of the Intermediate Value Theorem about my height?
3
Give an example (by sketch or formula) of a function \(f(x)\text{,}\) defined on the interval \([0,2]\text{,}\) with \(f(0)=0\text{,}\) \(f(2)=2\text{,}\) and \(f(x)\) never equal to 1. Why does this not contradict the Intermediate Value Theorem?
4
Is the following a valid statement?
Suppose \(f\) is a continuous function over \([10,20]\text{,}\) \(f(10)=13\text{,}\) and \(f(20)=-13\text{.}\) Then \(f\) has a zero between \(x=10\) and \(x=20\text{.}\)
5
Is the following a valid statement?
Suppose \(f\) is a continuous function over \([10,20]\text{,}\) \(f(10)=13\text{,}\) and \(f(20)=-13\text{.}\) Then \(f(15)=0\text{.}\)
6
Is the following a valid statement?
Suppose \(f\) is a function over \([10,20]\text{,}\) \(f(10)=13\text{,}\) and \(f(20)=-13\text{,}\) and \(f\) takes on every value between \(-13\) and \(13\text{.}\) Then \(f\) is continuous.
7
Suppose \(f(t)\) is continuous at \(t=5\text{.}\) True or false: \(t=5\) is in the domain of \(f(t)\text{.}\)
8
Suppose \(\ds\lim_{t \rightarrow 5}f(t)=17\text{,}\) and suppose \(f(t)\) is continuous at \(t=5\text{.}\) True or false: \(f(5)=17\text{.}\)
9
Suppose \(\ds\lim_{t \rightarrow 5}f(t)=17\text{.}\) True or false: \(f(5)=17\text{.}\)
10
Suppose \(f(x)\) and \(g(x)\) are continuous at \(x=0\text{,}\) and let \(h(x)=\dfrac{xf(x)}{g^2(x)+1}\text{.}\) What is \(\ds\lim_{x \to 0^+} h(x)\text{?}\)
Exercises — Stage 2
11
Find a constant \(k\) so that the function
\begin{equation*}
a(x)=\left\{\begin{array}{ll}
x\sin\left(\frac{1}{x}\right)&\mbox{when } x \neq 0\\
k&\mbox{when }x=0
\end{array}\right.
\end{equation*}
is continuous at \(x=0\text{.}\)
12
Use the Intermediate Value Theorem to show that the function \(f(x)=x^3+x^2+x+1\) takes on the value 12345 at least once in its domain.
13 (✳)
Describe all points for which the function is continuous: \(f(x)=\dfrac{1}{x^2-1}\text{.}\)
14 (✳)
Describe all points for which this function is continuous: \(f(x)=\dfrac{1}{\sqrt{x^2-1}}\text{.}\)
15 (✳)
Describe all points for which this function is continuous: \(\dfrac{1}{\sqrt{1+\cos(x)}}\text{.}\)
16 (✳)
Describe all points for which this function is continuous: \(f(x)=\dfrac{1}{\sin x}\text{.}\)
17 (✳)
Find all values of \(c\) such that the following function is continuous at \(x=c\text{:}\)
\begin{equation*}
f(x)=\left\{\begin{array}{ccc}
8-cx & \text{if} & x\le c\\
x^2 & \text{if} & x \gt c
\end{array}\right.
\end{equation*}
Use the definition of continuity to justify your answer.
18 (✳)
Find all values of \(c\) such that the following function is continuous everywhere:
\begin{align*}
f(x) &= \begin{cases}
x^2+c & x\geq 0\\
\cos cx & x \lt 0
\end{cases}
\end{align*}
Use the definition of continuity to justify your answer.
19 (✳)
Find all values of \(c\) such that the following function is continuous:
\begin{equation*}
f(x) = \begin{cases}
x^2-4 & \text{if } x \lt c\\
3x & \text{if } x \ge c\,.
\end{cases}
\end{equation*}
Use the definition of continuity to justify your answer.
20 (✳)
Find all values of \(c\) such that the following function is continuous:
\begin{equation*}
f(x)=\left\{\begin{array}{ccc}
6-cx & \text{if} & x\le 2c\\
x^2 & \text{if} & x \gt 2c
\end{array}\right.
\end{equation*}
Use the definition of continuity to justify your answer.
Exercises — Stage 3
21
Show that there exists at least one real number \(x\) satisfying \(\sin x = x-1\)
22 (✳)
Show that there exists at least one real number \(c\) such that \(3^c=c^2\text{.}\)
23 (✳)
Show that there exists at least one real number \(c\) such that \(2\tan(c)=c+1\text{.}\)
24 (✳)
Show that there exists at least one real number c such that \(\sqrt{\cos(\pi c)} = \sin(2 \pi c) + \frac{1}{2}\text{.}\)
25 (✳)
Show that there exists at least one real number \(c\) such that \(\dfrac{1}{(\cos\pi c)^2} = c+\dfrac{3}{2}\text{.}\)
26
Use the intermediate value theorem to find an interval of length one containing a root of \(f(x)=x^7-15x^6+9x^2-18x+15\text{.}\)
27
Use the intermediate value theorem to give a decimal approximation of \(\sqrt[3]{7}\) that is correct to at least two decimal places. You may use a calculator, but only to add, subtract, multiply, and divide.
28
Suppose \(f(x)\) and \(g(x)\) are functions that are continuous over the interval \([a,b]\text{,}\) with \(f(a) \leq g(a)\) and \(g(b)\leq f(b)\text{.}\) Show that there exists some \(c \in [a,b]\) with \(f(c)=g(c)\text{.}\)