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\(\newcommand{\Z}{\mathbb{Z}} \newcommand{\reals}{\mathbb{R}} \newcommand{\real}[1]{\mathbb{R}^{#1}} \newcommand{\fe}[2]{#1\mathopen{}\left(#2\right)\mathclose{}} \newcommand{\cinterval}[2]{\left[#1,#2\right]} \newcommand{\ointerval}[2]{\left(#1,#2\right)} \newcommand{\cointerval}[2]{\left[\left.#1,#2\right)\right.} \newcommand{\ocinterval}[2]{\left(\left.#1,#2\right]\right.} \newcommand{\point}[2]{\left(#1,#2\right)} \newcommand{\fd}[1]{#1'} \newcommand{\sd}[1]{#1''} \newcommand{\td}[1]{#1'''} \newcommand{\lz}[2]{\frac{d#1}{d#2}} \newcommand{\lzn}[3]{\frac{d^{#1}#2}{d#3^{#1}}} \newcommand{\lzo}[1]{\frac{d}{d#1}} \newcommand{\lzoo}[2]{{\frac{d}{d#1}}{\left(#2\right)}} \newcommand{\lzon}[2]{\frac{d^{#1}}{d#2^{#1}}} \newcommand{\lzoa}[3]{\left.{\frac{d#1}{d#2}}\right|_{#3}} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\sech}{\operatorname{sech}} \newcommand{\csch}{\operatorname{csch}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Activity2.11Continuity on an Interval

Now that we have a definition for continuity at a number, we can go ahead and define what we mean when we say a function is continuous over an interval.

Definition2.11.1Continuity on an Interval

The function \(f\) is continuous over an open interval if and only if it is continuous at each and every number on that interval.

The function \(f\) is continuous over a closed interval \(\cinterval{a}{b}\) if and only if it is continuous on \(\ointerval{a}{b}\text{,}\) continuous from the right at \(a\text{,}\) and continuous from the left at \(b\text{.}\) Similar definitions apply to half-open intervals.

Subsection2.11.1Exercises

1

Write a definition for continuity on the half-open interval \(\ocinterval{a}{b}\text{.}\)

Referring to the function in Figure 2.9.2, decide whether each of the following statements are true or false.

2

True or False? \(h\) is continuous on \(\cointerval{-4}{-1}\text{.}\)

3

True or False? \(h\) is continuous on \(\ointerval{-4}{-1}\text{.}\)

4

True or False? \(h\) is continuous on \(\ocinterval{-4}{-1}\text{.}\)

5

True or False? \(h\) is continuous on \(\ocinterval{-1}{2}\text{.}\)

6

True or False? \(h\) is continuous on \(\ointerval{-1}{2}\text{.}\)

7

True or False? \(h\) is continuous on \(\ointerval{-\infty}{-4}\text{.}\)

8

True or False? \(h\) is continuous on \(\ocinterval{-\infty}{-4}\text{.}\)

Several functions are described below. Your task is to sketch a graph for each function on its provided axis system. Do not introduce any unnecessary discontinuities or intercepts that are not directly implied by the stated properties. Make sure that you draw all implied asymptotes and label them with their equations.

9

\begin{align*} \fe{f}{0}=4\text{ and }\fe{f}{4}&=5\\ \lim_{x\to4^{-}}\fe{f}{x}=2\text{ and }\lim_{x\to4^{+}}\fe{f}{x}&=5\\ \lim_{x\to-\infty}\fe{f}{x}=\lim_{x\to\infty}\fe{f}{x}&=4 \end{align*}

\(f\) has at most one discontinuity.

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Figure2.11.2\(y=\fe{f}{x}\)
10

\begin{align*} \fe{g}{0}=4,\text{ }\fe{g}{3}=-2,\text{ and }\fe{g}{6}&=0\\ \lim_{x\to-2}\fe{g}{x}=\lim_{x\to-\infty}\fe{g}{x}&=\infty \end{align*}

\(g\) is continuous and has constant slope on \(\ointerval{0}{\infty}\text{.}\)

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Figure2.11.3\(y=\fe{g}{x}\)
11

\begin{align*} \fe{m}{-6}\amp=5\\ \lim_{x\to-4^{+}}\fe{m}{x}\amp=-2\\ \lim_{x\to3}\fe{m}{x}=\lim_{x\to\infty}\fe{m}{x}\amp=-\infty \end{align*}

\(m\) only has discontinuities at \(-4\) and \(3\text{.}\) \(m\) has no zeros. \(m\) has slope \(-2\) over \(\ointerval{-\infty}{-4}\text{.}\) \(m\) is continuous over \(\cointerval{-4}{3}\text{.}\)

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Figure2.11.4\(y=\fe{m}{x}\)