2.2.3How do limits and continuity relate?

Definition of Continuity

Math Notes

The Three Conditions of Continuity

Continuity is an important concept in calculus because many important theorems of calculus require functions to be continuous over the interval of interest. Simply stating that you can trace a graph without lifting your pencil is neither a complete nor a formal way to justify the continuity of a function at a point.

In order to justify that a function $f$ is continuous at the point
$x=a$ , you must demonstrate that $f$ meets all three conditions listed below.

$1. \lim \limits_{x \rightarrow a} f(x) \text{ exists} \\ 2. f(a) \text{ exists}\\ 3. \lim \limits_{x \rightarrow a} f(x) =f(a)$

Speech bubble pointing to item, limit as x approaches A, of f of x exists: Recall that this means for a function, the limit as x approaches A, from the negative is the same as the limit as x approaches A, from the positive.

A function is continuous over an interval if it is continuous at each point in the interval.

Continuous curve, coming from lower left, turning down, then up, both in first quadrant, continuing up & right, with 3 tick marks on x axis, center one labeled, a, aligned with second turning point.

2-73.

Examine the conditions of continuity given in the Math Notes box above and summarize them with your team. Then demonstrate your understanding of continuity by sketching functions for parts (a) through (c).

Sketch a function that satisfies condition 1, but not condition 2 (and therefore not condition 3).
Sketch a function that satisfies condition 2, but not conditions 1 or 3.
Sketch a function that satisfies conditions 1 and 2, but not condition 3.

2-74.

Examine the graph at right. Identify four values of $x$ where a discontinuity exists. At each of these values, state the condition(s) of continuity that fail(s).

2-75.

If $g$ is continuous for all real numbers, such that $g(–4) = -10$ and $g(-1) = 3$ , explain why $g$ must have a root ( $x$ -intercept) for an $x$ -value in the interval $(–4, -1)$ . Include a sketch of a possible function.

2-76.

Use the three conditions of continuity to justify why $f(x) = \left | x \right |$ is continuous at $x = 0$ .

2-77.

While waiting for a bus, you and your friends see a car traveling at $65$ mph. When the driver notices you, he instantly slams on the brakes and comes to a stop.

True of false: While breaking, the driver traveled at every intermediate speed between $65$ mph and $0$ mph.
Must the graph of this situation be a continuous function?

2-78.

Explain why a function that is continuous for all $x$ -values on $[a, b]$ must pass through every $y$ -value between $f (a)$ and $f (b)$ at least once in that interval. This is called the Intermediate Value Theorem for continuous functions.

Continuous curve, coming from lower left, turning down in second quadrant, turning up in first quadrant, continuing up & right, with a highlighted point on the curve in second quadrant, with horizontal & vertical dashed lines to each axis, labeled a, on x axis, & a highlighted point in first quadrant with horizontal & vertical dashed lines to each axis labeled, b, on x axis.

2-79.

Examine the function shown below. Notice that $f (–2) = –2$ and $f (2) = 2$ , yet there is no root between $x = –2$ and $2$ . Why does this not contradict the Intermediate Value Theorem?

2 rays, left starting at open point (0, comma negative 1), passing through (negative 2, comma negative 2), continuing down & left, right starting at closed point (0, comma 1), passing through (2, comma 2), continuing up & right.

2-80.

For some continuous function $f, f (–3) = 5$ and $f (2) = -3$ . What is the minimum number of values possible for $a$ that satisfy $f(a) = 1$ ?

2-81.

A helium balloon is released from the ground and floats upward. The height of the balloon is shown at the following times: Homework Help ✎

Time (s)	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
Height (feet)	$0$	$50$	$98$	$144$	$188$	$230$	$270$	$308$	$344$	$378$	$410$

What is the average velocity over the first $10$ seconds of the balloon’s flight? Over the first $5$ seconds?
Calculate the finite differences for the heights. How is the velocity changing? Explore this using the 2-81 HW eTool.
What do the finite differences tell you about the height function for the balloon?

2-82.

Examine the expanded sums below and write the equivalent sigma notation. Homework Help ✎

$\frac { 2 } { 3 } f ( - 2 + \frac { 2 } { 3 } \cdot 0 ) + \frac { 2 } { 3 } f ( - 2 + \frac { 2 } { 3 } \cdot 1 ) + \frac { 2 } { 3 } f ( - 2 + \frac { 2 } { 3 } \cdot 2 ) + \frac { 2 } { 3 } f ( - 2 + \frac { 2 } { 3 } \cdot 3 )$
$\frac { 1 } { 2 } f ( 6 + \frac { 1 } { 2 } \cdot 0 ) + \frac { 1 } { 2 } f ( 6 + \frac { 1 } { 2 } \cdot 1 ) + \frac { 1 } { 2 } f ( 6 + \frac { 1 } { 2 } \cdot 2 ) + \frac { 1 } { 2 } f ( 6 + \frac { 1 } { 2 } \cdot 3 ) + \frac { 1 } { 2 } f ( 6 + \frac { 1 } { 2 } \cdot 4 )$

2-83.

The Intermediate Value Theorem is sometimes used to prove that roots exist. For example, $f(x) =5 \sqrt [ 3 ] { x - 2 } -4$ is a continuous function. Given $f (2) = -4$ and $f (3) = 1$ , does $f$ have a root somewhere between $x = 2$ and $x = 3$ ? Why or why not? 2-83 HW eTool. Homework Help ✎

2-84.

Write a Riemann sum for a general function $f$ to estimate the area under the curve for $2\le x\le5$ using $n$ left endpoint rectangles of equal width if: Homework Help ✎

$n = 3$ rectangles

$n = 9$ rectangles

$n = 300$ rectangles

2-85.

Jamal wrote the following Riemann sum to estimate the area under $f (x) = 3x^2 - 2$ . 2-85 HW eToo. Homework Help ✎

$\displaystyle \sum _ { i = 0 } ^ { 9 } \frac { 1 } { 2 } f ( - 3 + \frac { 1 } { 2 } i )$

Draw a sketch of the region. How many rectangles did he use?
For what domain of $f$ did Jamal estimate the area?
Use the summation feature of your calculator to approximate the area using Jamal’s Riemann sum.

2-86.

The manager of Books-To-Go knows that the rate of daily sales (in books per day) varies over the course of a week. This rate can be represented by the step function shown in the graph below. Using this data, calculate how many books this store sold during this week. What is the average number of books sold per day? Homework Help ✎

2-87.

For each description below, write a limit equation and sketch a possible function. Homework Help ✎

As $x \rightarrow 0, y \rightarrow 9$ .
As $x$ gets closer to $3$ on both sides, $f (x)$ becomes increasingly large.
Which of the limits from parts (a) and (b) exist? Explain your reasoning.

2-88.

Given $f ( x ) = \left\{ \begin{array} { c l } { x ^ { 2 } } & { \text { for } x < 1 , \text { but } x \neq - 1 } \\ { 3 } & { \text { for } x = 1 } \\ { 2 x - 1 } & { \text { for } 1 < x < 3 } \\ { 4 } & { \text { for } x \geq 3 } \end{array} \right.$ 2-88 HW eTool. Homework Help ✎

Sketch $y = f (x)$ .
For what values of $x$ is $f$ not continuous?

Calculus Third Edition

The Three Conditions of Continuity

Intermediate Value Theorem