## Exercises1.3Continuity

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

1. The function $f$ is defined at the number $a\text{;}$

2. The limit $\displaystyle{\lim_{x\to a}}f(x)$ exists;

3. $\displaystyle{\lim_{x\to a}f(x)=f(a)}\text{.}$

###### 1.

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*}
1. Find the value of the constant $c$ so that the function $f(x)$ is continuous.

2. For the value of $c$ found in part (a), verify whether the 3 conditions for continuity are satisfied.

3. Draw a graph of $f(x)$ from $x=-\pi$ to $x=3\pi$ indicating the scaling used.

Hint

To find $c \text{,}$ solve $\displaystyle \lim _{x\to \pi ^-}f(x)=\lim _{x\to \pi ^-}f(x)$ for $c\text{.}$

1. $c=\pi\text{.}$
2. For all $x \le \pi \text{,}$ $f = \pi - x$ is a linear function and so is continuous. For all $x \ge \pi \text{,}$ $f = \pi \sin x$ is a sine function and so is continuous. When $x = \pi \text{,}$ we have:

1. $f(\pi) = 0 \text{.}$

2. $\displaystyle \lim _{x\to \pi ^-} \left(\pi - x \right) = 0 \text{,}$ and $\displaystyle \lim _{x\to \pi ^-} \pi \sin(x) = 0\text{.}$ Therefore, $\displaystyle \lim _{x\to \pi} f(x)$ exists, and is equal to $0\text{.}$

3. $\displaystyle \lim _{x\to \pi} f(x) = 0 = f(\pi) \text{.}$

###### 2.
1. Use the Intermediate Value Theorem to show that $\displaystyle 2^x=\frac{10}{x}$ for some $x>0\text{.}$

2. Show that the equation $\displaystyle 2^x=\frac{10}{x}$ has no solution for $x\lt 0\text{.}$

Solution
1. Let $\displaystyle f(x)=2^x-\frac{10}{x}\text{.}$ Note that the domain of $f$ is the set $\mathbb{R}\backslash \{ 0\}$ and that on its domain, as a sum of two continuous function, $f$ is continuous.

2. For all $x\in (-\infty ,0)$ we have that $\displaystyle \frac{10}{x}\lt 0$ which implies that for all $x\in (-\infty ,0)$ we have that all $f(x)>0\text{.}$

###### 3.

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 YES or NO:

1. Is $f$ continuous at:

1. $x=1\text{.}$

2. $x=6\text{.}$

2. Do the following limits exist?

1. $\displaystyle \lim _{x\to 1}f(x)\text{.}$

2. $\displaystyle \lim _{x\to 3^-}f(x)\text{.}$

3. Is $f$ differentiable

1. at $x=1\text{?}$

2. on $(1,3)\text{?}$

1. No.

2. Yes.

1. Yes.

2. Yes.

1. No.

2. No.

###### 4.

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*}
1. Determine whether or not $f$ is continuous at $x=1\text{.}$ Justify your answer and state your conclusion.

2. Using the definition of the derivative, determine $f'(1)\text{.}$

1. Check that $\displaystyle \lim _{x\to 1^-}f(x)=\lim _{x\to 1^+}f(x)=f(1)\text{.}$

2. $\displaystyle \frac{1}{2}\text{.}$ Note $\displaystyle \lim _{x\to 1^-}\frac{\frac{5+x}{2}-3}{x-1}=\frac{1}{2}$ and $\displaystyle \lim _{x\to 1^+}\frac{(2+\sqrt{x})-3}{x-1}=\frac{1}{2}\text{.}$

###### 5.
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*}
$-1\text{.}$
###### 6.
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.
$\sqrt[3]{9}\text{.}$
###### 7.
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.
$\ds \frac{2}{3}\text{.}$
###### 8.
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*}
$\ds -\frac{1}{\pi}\text{.}$
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.
$\displaystyle f(x)=\frac{x^2-9}{x-3}$ if $x\not= 3$ and $f(3)=0\text{.}$
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 $c\in(0,1)\text{.}$
Consider the function $g(x)=f(x)-x\text{.}$