Free Practice Quiz Question List

03 Continuity Online Quiz Questions

Use this free practice quiz with 20 questions to review 03 Continuity, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

True or false: If a function is differentiable at x=a, then it is continuous at x=a.

A
True
B
False

True or false

1 point

True or false: Because f(x)=1/x has opposite signs at x=-1 and x=1, the Intermediate Value Theorem guarantees a solution to 1/x=0 in (-1,1).

A
True
B
False

Choose one

1 point

Which statement gives all three conditions required for a function f to be continuous at x=a?

A
The limit must exist, but f(a) may be undefined.
B
f(a) must be defined, the limit must exist, and the limit must equal f(a).
C
The function must be differentiable at a.
D
The left- and right-hand limits must be infinite.

Choose one

1 point

What type of discontinuity does f(x)=(x^2-9)/(x-3) have at x=3?

A
A jump discontinuity
B
An infinite discontinuity
C
A removable discontinuity
D
No discontinuity

Choose one

1 point

Let f(u)=1/(u-3) and g(x)=x^2. On which set is the composite (f∘g)(x) continuous?

A
All real numbers
B
All real numbers except x=3
C
All real numbers except x=-√3 and x=√3
D
Only x=-√3 and x=√3

Choose all

1 point

Select all conditions that are part of continuity on a closed interval [a,b].

A
Continuity at every point in the open interval (a,b)
B
Two-sided continuity from outside the interval at both endpoints
C
Right-hand continuity at a
D
Left-hand continuity at b
E
Differentiability at every point of [a,b]

Choose all

1 point

Suppose f and g are continuous at x=a. Select all expressions that are necessarily continuous at x=a under the stated information.

A
The sum f+g
B
The difference f-g
C
The quotient f/g regardless of g(a)
D
The product fg
E
Every composition f∘g without any domain condition

Written response

1 point

For f(x)=x^2+3x-1, what is the value of f(2)?

Written response

1 point

What happens to the limit of f(x)=sin(1/x) as x approaches 0? Enter a brief term.

Fill in the blank

1 point

Complete the key equality for continuity at x=a: provided f(a) is defined, .

Fill in the blank

1 point

Complete the root-existence conclusion: If f is continuous on [a,b] and f(a)f(b)<0, then .

Open ended

1 point

Use the Intermediate Value Theorem to prove that x^3+x-1=0 has at least one solution in the interval (0,1).

Choose one

1 point

What is the inverse function of f(x)=3x-5?

A
f^-1(x)=3x+5
B
f^-1(x)=(x-5)/3
C
f^-1(x)=(x+5)/3
D
f^-1(x)=5-3x

True or false

1 point

True or false: If a function is differentiable at $x=a$ , then it must be continuous at $x=a$ .

A
True
B
False

Choose one

1 point

Suppose $f$ is defined on the closed interval $[a,b]$ . Which condition is required for continuity at the left endpoint $a$ ?

A
$\lim_{x\to a^-}f(x)=f(a)$
B
$\lim_{x\to a^+}f(x)=f(a)$
C
Both two-sided limits must exist outside the interval
D
No endpoint condition is required

Choose one

1 point

Let $f(x)=x^2+3x-1$ . What is $\lim_{x\to 2}f(x)$ ?

A
$5$
B
$7$
C
$9$
D
$11$

Written response

1 point

For $f(x)=x^2+3x-1$ , enter the value of $f(2)$ .

Choose one

1 point

Consider $f(x)=0$ for $x<1$ and $f(x)=2$ for $x\ge 1$ . What type of discontinuity occurs at $x=1$ ?

A
Jump discontinuity
B
Removable discontinuity
C
Infinite discontinuity
D
No discontinuity

Written response

1 point

For $f(x)=\frac{x^2-4}{x-2}$ , what standard term describes the discontinuity at $x=2$ ? Enter the term only.

Choose one

1 point

A student claims that the Intermediate Value Theorem proves $1/x=0$ has a solution in $[-1,1]$ because $f(-1)=-1$ and $f(1)=1$ . What is the flaw in this reasoning?

A
The endpoint values do not have opposite signs
B
The function is not continuous on $[-1,1]$
C
The interval is closed
D
The target value $0$ is not between the endpoint values