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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 \(\lim\limits_{x\to a}\fe{f}{x}=\fe{f}{a}\).

There are three ways that the defining property can fail to be satisfied at a given value of \(a\). 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 \(a\), then Property 3 also fails to be satisfied at \(a\).

Subsection2.9.1Exercises

These questions refer to the function in Figure 2.9.2.

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.