Lesson test: Continuity

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Continuity

1.[2p]
Continuity of $f$ at $a$ is the single equation $lim_{x \to a} f (x) = f (a)$ . What three demands does it make?

The value exists, the limit exists, and they are equal

Correct

The function is defined on both sides, is bounded, and has a derivative

The answer is: The value exists, the limit exists, and they are equal

The limit exists, the function is increasing, and the graph has no corner

The answer is: The value exists, the limit exists, and they are equal

The value exists, the function is bounded, and the graph can be drawn

The answer is: The value exists, the limit exists, and they are equal

2.[2p]
The function equals $(x^{2} - 25) / (x - 5)$ for $x \neq 5$ and equals $k$ at $x = 5$ . What value of $k$ makes it continuous?

CorrectNot quite: 10

3.[3p]
The function equals $x^{2}$ for $x \leq 2$ and $c x + 1$ for $x > 2$ . What value of $c$ makes it continuous everywhere?

CorrectNot quite: 1.5

4.[2p]
What kind of discontinuity does $\frac{x^{2} - x - 6}{x^{2} - 9}$ have at $x = 3$ ?

Removable, since the limit exists and only the value is missing

Correct

Infinite, since the denominator vanishes there

The answer is: Removable, since the limit exists and only the value is missing

A jump, since the one-sided limits disagree

The answer is: Removable, since the limit exists and only the value is missing

None, since the function is continuous there

The answer is: Removable, since the limit exists and only the value is missing

5.[3p]
Over the rational numbers alone, a continuous function that changes sign on an interval must still have a root in it.

True

The answer is: False

False

Correct

6.[2p]
Bisection is applied to $f (x) = x^{3} - x - 1$ starting from $[1, 2]$ . Given $f (1.5) = 0.875$ , which point is tested next?

CorrectNot quite: 1.25

7.[3p]
Which of these guarantee that a continuous function attains a maximum value?

Select all that apply

The domain is a closed bounded interval

Correct

The domain is the open interval $(0, 1)$

The answer is: The domain is a closed bounded interval, The domain is $[0, 1]$

The domain is $[0, 1]$

Correct

The function is bounded on an open interval

The answer is: The domain is a closed bounded interval, The domain is $[0, 1]$

8.[3p]
Match each function to how continuity fails at the point named.

$(x^{2} - 9) / (x - 3)$ at $x = 3$
$1 / x$ at $x = 0$
A postage rate at a weight limit
$\sin (1 / x)$ at $x = 0$

a hole, repairable by one value
endless oscillation, no limit
unbounded, not repairable
a jump between two one-sided limits

Show the answer

$(x^{2} - 9) / (x - 3)$ at $x = 3$ : a hole, repairable by one value $1 / x$ at $x = 0$ : unbounded, not repairable A postage rate at a weight limit: a jump between two one-sided limits $\sin (1 / x)$ at $x = 0$ : endless oscillation, no limit

9.[2p]
Why does the Weierstrass function of 1872 matter?

It is continuous everywhere and has a derivative nowhere, ending the assumption that unbroken curves have tangents

Correct

It is the first function shown to be discontinuous at every point

The answer is: It is continuous everywhere and has a derivative nowhere, ending the assumption that unbroken curves have tangents

It proves the Intermediate Value Theorem

The answer is: It is continuous everywhere and has a derivative nowhere, ending the assumption that unbroken curves have tangents

It shows that continuous functions need not be bounded

The answer is: It is continuous everywhere and has a derivative nowhere, ending the assumption that unbroken curves have tangents

1.[2p] Continuity of f at a is the single equation limx→af(x)=f(a). What three demands does it make?

2.[2p] The function equals (x2-25)/(x-5) for x≠5 and equals k at x=5. What value of k makes it continuous?

3.[3p] The function equals x2 for x≤2 and cx+1 for x>2. What value of c makes it continuous everywhere?

4.[2p] What kind of discontinuity does x2-x-6x2-9 have at x=3?

5.[3p] Over the rational numbers alone, a continuous function that changes sign on an interval must still have a root in it.

6.[2p] Bisection is applied to f(x)=x3-x-1 starting from [1,2]. Given f(1.5)=0.875, which point is tested next?

7.[3p] Which of these guarantee that a continuous function attains a maximum value?

8.[3p] Match each function to how continuity fails at the point named.

9.[2p] Why does the Weierstrass function of 1872 matter?

1.[2p]
Continuity of $f$ at $a$ is the single equation $lim_{x \to a} f (x) = f (a)$ . What three demands does it make?

2.[2p]
The function equals $(x^{2} - 25) / (x - 5)$ for $x \neq 5$ and equals $k$ at $x = 5$ . What value of $k$ makes it continuous?

3.[3p]
The function equals $x^{2}$ for $x \leq 2$ and $c x + 1$ for $x > 2$ . What value of $c$ makes it continuous everywhere?

4.[2p]
What kind of discontinuity does $\frac{x^{2} - x - 6}{x^{2} - 9}$ have at $x = 3$ ?

5.[3p]
Over the rational numbers alone, a continuous function that changes sign on an interval must still have a root in it.

6.[2p]
Bisection is applied to $f (x) = x^{3} - x - 1$ starting from $[1, 2]$ . Given $f (1.5) = 0.875$ , which point is tested next?

7.[3p]
Which of these guarantee that a continuous function attains a maximum value?

8.[3p]
Match each function to how continuity fails at the point named.

9.[2p]
Why does the Weierstrass function of 1872 matter?