$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

1.6E Excercises

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Exercise $$\PageIndex{1}$$: Terms and Concepts

1. In your own words, describe what it means for a function to be continuous.
2. In your own words, describe what the Intermediate Value Theorem states.
3. What is a “root” of a function?
4. T/F: If $$f$$ is defined on an open interval containing c, and $$\lim\limits_{x\to c} f(x)$$ exists, then $$f$$ is continuous at c.
5. T/F: If $$f$$ is continuous at c, then $$\lim\limits_{x\to c} f(x)$$ exist T/F: If $$f$$ is continuous at c, then $$\lim\limits_{x\to c^+} f(x)=f(c)$$.
6. T/F: If $$f$$ is continuous on [a, b], then $$\lim\limits_{x\to a^-} f(x)=f(a)$$.
7. T/F: If f is continuous on [0, 1) and [1, 2), then $$f$$ is continuous on [0, 2).
8. T/F: The sum of continuous functions is also continuous.

Add answer text here and it will automatically be hidden if you have a "AutoNum" template active on the page.

Exercise $$\PageIndex{2}$$: Continuity

Determine on which interval(s) the given function is continuous.

1. $$f(x)=x^2-3x+9$$
2. $$g(x) = \sqrt{x^2-4}$$
3. $$h(k) = \sqrt{1-k}+\sqrt{k+1}$$
4. $$f(t) = \sqrt{5t^2-30}$$
5. $$g(t) = \frac{1}{\sqrt{1-t^2}}$$
6. $$g(x) = \frac{1}{1+x^2}$$
7. $$f(x) = e^x$$
8. $$g(s) = \ln s$$
9. $$h(t) = \cos t$$
10. $$f(k) = \sqrt{1-e^k}$$
11. $$f(x) = \sin (e^x+x^2)$$

Add answer text here and it will automatically be hidden if you have a "AutoNum" template active on the page.

Exercise $$\PageIndex{3}$$: Continuity

For what values of $$x$$, is $$f$$ continuous:

1. Given $$f(x)= \left\{ \begin{array}{ccc} \displaystyle \frac{x^2-5x+6}{3-x} & \mbox{ if } x <3\\ \\ -1 & \mbox{ if } x =3 \\ \\ x-2 & \mbox{ if } x >3 \\ \end{array} \right.$$
2. Given $$f(x)= \left\{ \begin{array}{ccc} \displaystyle \frac{-6}{x} & \mbox{ if } x \leq -1\\ \\ 3x^2 & \mbox{ if } x >-1 \\ \end{array} \right.$$
3. Given $$f(x)= \left\{ \begin{array}{ccc} \displaystyle \frac{|x-3|-2}{x(x-1)}& \mbox{ if } x \neq 0,x \neq 1\\ \\ 1& \mbox{ if } x =1 \\ \\ 1& \mbox{ if } x =0 \\ \end{array} \right.$$

$$\mathbb{R}\setminus\{3\},\(\mathbb{R}\setminus\{-1\}, \(\mathbb{R}\setminus\{0,1\}$$

Exercise $$\PageIndex{4}$$:Continuity

For what values of $$k$$, each of the following function is continuous every where:

1. $$f(x)= \left\{ \begin{array}{ccc} (x+k)^2 & \mbox{ if } x\geq 0\\ \\ (2-x)^2 & \mbox{ if } x <0 \\ \end{array} \right.$$
2. $$f(x)= \left\{ \begin{array}{ccc} (x+k) & \mbox{ if } x\leq 2\\ \\ \sqrt{kx^2+20} & \mbox{ if } x >2 \\ \end{array} \right.$$
3. $$f(x) = \left\{ \begin{array}{l} 2x+k, x < 3\\ \\ \displaystyle \frac{kx^2 + 1}{7}, x \geq 3\end{array}\right .$$

4. $$f(x) = \left\{ \begin{array}{cc} x^2+kx+1 & \mbox{if x \leq 1,} \\ \\ 2x+3 & \mbox{if x > 1.} \end{array} \right.$$

$$k=\pm 2, k=\pm 4$$

Exercise $$\PageIndex{5}$$:Continuity

Given $$f(x)= \left\{ \begin{array}{ccc} \displaystyle \frac{\sin(2x)}{ax}& \mbox{ if } x < 0,\\ \\ x^2-5& \mbox{ if } 0 \leq x <2 \\ \\ bx-4& \mbox{ if } x \geq 2 \\ \end{array} \right.$$

1. For what value of $$a$$, $$f$$ is continuous at $$x=0$$.
2. For what value of $$b$$, $$f$$ is continuous at $$x=2$$.

$$a=-10, b=3/2$$

Exercise $$\PageIndex{6}$$: Continuity

Find values of $$k$$ and $$m,$$ if possible, that will make the function $$f$$ continuous everywhere.
$$f(x) = \left\{ \begin{array}{ll} x^2+2, x>1\\m(x+1)+k, & - 1< x \leq 1\\ 2x^2 + x + 5, & x \leq -1. \end{array}\right.$$

$$k=6 , m=-3/2$$

Solution:

when $$x=1$$:

$$\lim_{x \to 1^+} f(x)= \lim_{x \to 1} x^2+2=3$$ and

$$\lim_{x \to 1^-} f(x)= \lim_{x \to 1} m(x+1)+k=2m+k=f(1)$$,

Since $$f$$ is continuous at $$x=1$$, $$2m+k=3$$.

when $$x=-1$$:

$$\lim_{x \to -1^+} f(x)= \lim_{x \to -1} m(x+1)+k=k$$ and

$$\lim_{x \to -1^-} f(x)= \lim_{x \to -1} 2x^2+x+5=2-1+5=6=f(1)$$,

Since $$f$$ is continuous at $$x=-1$$, $$k=6$$.

Therefore, $$2m+6=3$$, implies $$m=-3/2$$.

Exercise $$\PageIndex{7}$$: Intermediate Value Theorem

Use the Intermediate Value Theorem to show that the equation $$x^3+x^2-2x-1=0$$ has at least one solution in the interval $$[1,3]$$.

Let $$f(x)= x^3+x^2-2x-1$$. Then $$f(x)$$ is continuous and $$f(1)=-1$$ and $$f(3)=29$$.
Since $$-1<0<29$$,by the Intermediate Value Theorem there exist at least one real number $$k$$ in the interval $$[1,3]$$ such that $$f(k)=0$$.