CLM Continuity

Activity 2.9 Continuity

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.

Definition 2.9.1. Continuity.

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.

\(\fe{f}{a}\) must be defined
\(\lim\limits_{x\to a}\fe{f}{x}\) must exist
\(\lim\limits_{x\to a}\fe{f}{x}\) 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 \(a\text{.}\)

Subsection Exercises

These questions refer to the function in Figure 2.9.2.

a plot of a function where the horizontal axis is labeled with t, and the vertical with y; one segment of the plot is a straght line with slope 1 coming in from the left, upward and to the right until reaching the point (-4,1); there is a hollow dot at (-4,1); the next segment of the curve starts at (-4,4) with a solid dot; the curve moves to the right in a stratight, flat line, until reaching (-2,4); at this point, it sharply bends to having slope -1, and extends down to (3,1); along the way, there is a hoollow dot at (-1,3) and at (2,0), and a solid dot hovering below the curve at (2-3); picking up back at (3,-1), the curve bends sharply upward and curves steeper and steeper until it exits the plot a little to the left of the vertical line t=5; a third segment of the curve comes back in from the top, a little to the right of the line t=5; it moves downward and slightly to the right; around (6,1), it begins moving more rightward than downward, and exits the plot on the right a little above the t-axis. — Figure 2.9.2. \(y=\fe{h}{t}\)

1.

Complete Table 2.9.3.

\(a\)	\(\fe{h}{a}\)	\(\lim\limits_{t\to a^{-}}\fe{h}{t}\)	\(\lim\limits_{t\to a^{+}}\fe{h}{t}\)	\(\lim\limits_{t\to a}\fe{h}{t}\)
\(-4\)	\(\phantom{\text{the answer}}\)	\(\phantom{\text{the answer}}\)	\(\phantom{\text{the answer}}\)	\(\phantom{\text{the answer}}\)
\(-1\)
\(2\)
\(3\)
\(5\)

Table 2.9.3. Function 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.