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Exercises 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

  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 \(c\text{.}\)

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

  3. Figure 1.1. \(c=\pi\)
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 \((1,3)\text{?}\)

Answer

    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 \(f'(1)\text{.}\)

Answer

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

Figure 1.2. A continuous function
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*}
is continuous everywhere. Justify your answer.
Answer
\(-1\text{.}\)
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.
Answer
\(\sqrt[3]{9}\text{.}\)
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.
Answer
\(\ds \frac{2}{3}\text{.}\)
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*}
is continuous everywhere. Justify your answer.
Answer
\(\ds -\frac{1}{\pi}\text{.}\)
9.
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.
Answer
\(\displaystyle f(x)=\frac{x^2-9}{x-3}\) if \(x\not= 3\) and \(f(3)=0\text{.}\)
10.
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.
Answer
11.
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.
Answer
12.
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 \(c\in(0,1)\text{.}\)
Hint
Consider the function \(g(x)=f(x)-x\text{.}\)