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

Activity2.9Continuity

Many statements we make about functions are only true over intervals where the function is continuous. When we say a function is continuous over an interval, we basically mean that there are no breaks in the function over that interval; that is, there are no vertical asymptotes, holes, jumps, or gaps along that interval.

Definition2.9.1Continuity

The function \(f\) is continuous at the number \(a\) if and only if \(\lim\limits_{x\to a}\fe{f}{x}=\fe{f}{a}\text{.}\)

There are three ways that the defining property can fail to be satisfied at a given value of \(a\text{.}\) To facilitate exploration of these three manners of failure, we can separate the defining property into three sub-properties.

  1. \(\fe{f}{a}\) must be defined

  2. \(\lim\limits_{x\to a}\fe{f}{x}\) must exist

  3. \(\lim\limits_{x\to a}\fe{f}{x}\) must equal \(\fe{f}{a}\)

Please note that if either Property 1 or Property 2 fails to be satisfied at a given value of \(a\text{,}\) then Property 3 also fails to be satisfied at \(a\text{.}\)

Subsection2.9.1Exercises

These questions refer to the function in Figure 2.9.2.

<<SVG image is unavailable, or your browser cannot render it>>

Figure2.9.2\(y=\fe{h}{t}\)
1

Complete Table 2.9.3.

\(a\) \(\fe{h}{a}\) \(\lim\limits_{t\to a^{-}}\fe{h}{t}\) \(\lim\limits_{t\to a^{+}}\fe{h}{t}\) \(\lim\limits_{t\to a}\fe{h}{t}\)
\(-4\) \(\phantom{\text{the answer}}\) \(\phantom{\text{the answer}}\) \(\phantom{\text{the answer}}\) \(\phantom{\text{the answer}}\)
\(-1\)
\(2\)
\(3\)
\(5\)
Table2.9.3Function values and limit values for \(h\)
2

State the values of \(t\) at which the function \(h\) is discontinuous. For each instance of discontinuity, state (by number) all of the sub-properties in Definition 2.9.1 that fail to be satisfied.