Continuity of Functions: Interactive Learning Quizzes for Advanced Mathematics

Question 1

Which of the following is a necessary condition for a function to be continuous at a point?

Accepted Answer

The limit of the function at that point exists and is equal to the value of the function at that point.

Answer

The value of the function at that point exists.

Answer

The derivative of the function at that point exists.

Answer

The function is differentiable at that point.

Question 2

If f(x) is continuous on [a, b] and g(x) is continuous on [b, c], then the function h(x) = f(x) + g(x) is continuous on:

Accepted Answer

[a, c]

Answer

[a, b]

Answer

[b, c]

Answer

[a - b, c - b]

Question 3

Which of the following theorems guarantees that a continuous function on a closed interval [a, b] has a maximum and minimum value?

Accepted Answer

Extreme Value Theorem

Answer

Intermediate Value Theorem

Answer

Rolle's Theorem

Answer

Mean Value Theorem

Question 4

Which of the following functions is NOT continuous at x = 0?

Accepted Answer

f(x) = 1 / x

Answer

f(x) = x^2

Answer

f(x) = |x|

Answer

f(x) = e^x

Question 5

Which of the following is NOT a property of continuous functions?

Accepted Answer

Their derivatives exist everywhere.

Answer

They can be graphed without any breaks or jumps.

Answer

Their limits exist at every point in their domain.

Answer

They are integrable.

Question 6

Which of the following theorems states that if the limit of f(x) as x approaches c exists and is equal to f(c), then f(x) is continuous at x = c?

Accepted Answer

Continuity Theorem

Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Question 7

Which of the following is the equation of a vertical asymptote to the graph of the function f(x) = (x^2 - 4)/(x - 2)?

Accepted Answer

x = 2

Answer

y = 2

Answer

x = -2

Answer

y = 4

Question 8

Can a function be continuous at a point but not differentiable at that point?

Accepted Answer

Yes

Answer

Only if the function has a horizontal asymptote at that point.

Answer

No

Answer

Only if the function has a vertical asymptote at that point.

Question 9

Determine if the function f(x) = (x^2 - 1)/(x - 1) is continuous at x = 1.

Accepted Answer

Not continuous

Answer

Continuous from the right only

Answer

Continuous from the left only

Answer

Continuous

Question 10

Which of the following is a necessary condition for a function to be continuous on an interval?

Accepted Answer

Limit of the function as x approaches any point in the interval exists.

Answer

The function must be increasing on the interval.

Answer

The function must have a finite derivative on the interval.

Question 11

Which of the following functions has an infinite discontinuity at x = 1?

Accepted Answer

f(x) = 1/(x - 1)

Answer

f(x) = x^2 - 1

Answer

f(x) = |x - 1|

Answer

f(x) = sin(x - 1)

Question 12

Determine if the function f(x) = { x^2 if x < 1, 2x - 1 if x >= 1 } is continuous at x = 1.

Accepted Answer

Not continuous

Answer

Continuous

Answer

Continuous from the right only

Answer

Continuous from the left only

Question 13

Which of the following is a property of continuous functions?

Accepted Answer

Their graphs can be sketched without lifting the pen.

Answer

They are always bounded.

Answer

They are always increasing.

Answer

They are always differentiable.

Question 14

Which of the following is a consequence of a function being continuous on a closed interval?

Accepted Answer

The function must attain both its maximum and minimum values on the interval.

Answer

The function must be differentiable on the interval.

Answer

The function must be increasing on the interval.

Question 15

Which of the following statements about continuous functions is true?

Accepted Answer

Continuous functions can be graphed without lifting the pen.

Answer

Continuous functions have no critical points.

Answer

Continuous functions are always increasing.

Answer

Continuous functions are always differentiable.

Question 16

Which of the following is an example of a function that is continuous at every point but not differentiable at some points?

Accepted Answer

f(x) = |x|

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Answer

f(x) = x^2

Question 17

What is the definition of continuity for a function f(x) at a point c?

Accepted Answer

f(c) is defined, the limit of f(x) as x approaches c exists, and this limit is equal to f(c).

Answer

The limit of f(x) as x approaches c exists and is equal to f(c)

Answer

f(c) is defined and the limit of f(x) as x approaches c exists

Question 18

Is the function f(x) = |x - 2| continuous at x = 2?

Accepted Answer

Yes

Answer

No, because the left-hand limit and the right-hand limit are not equal

Answer

No, because the function is not defined at x = 2

Question 19

Identify the points of discontinuity of the function f(x) = (x - 1) / (x^2 - 1).

Accepted Answer

x = ±1

Answer

The function is continuous at all real numbers

Answer

x = 0 and x = 2

Answer

x = -2 and x = 2

Question 20

Which of the following functions is not differentiable at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = sin(x)

Answer

f(x) = x^2

Answer

f(x) = e^x

Question 21

Which of the following is NOT a necessary condition for a function to be continuous on an interval?

Accepted Answer

The function must be differentiable on the interval

Answer

The function must be bounded on the interval

Answer

The function must have a finite number of discontinuities on the interval

Question 22

Identify the function that is not continuous everywhere.

Accepted Answer

f(x) = 1 / (x - 1)

Answer

f(x) = e^x

Answer

f(x) = x^3

Question 23

Which property is NOT applicable to continuous functions?

Accepted Answer

Continuous functions must be differentiable.

Answer

Continuous functions have no abrupt changes in their graphs.

Answer

Continuous functions can be integrated.

Answer

Continuous functions are defined at all points in their domain.

Question 24

Identify the theorem used to determine the continuity of a composite function.

Accepted Answer

Chain Rule for Continuity

Answer

Squeeze Theorem

Answer

Mean Value Theorem

Answer

Intermediate Value Theorem

Question 25

Evaluate whether the function f(x) = 1/x is continuous at x = 0:

Accepted Answer

The function is not continuous at x = 0

Answer

The function is continuous everywhere

Answer

The function is continuous only for x < 0

Answer

The function is continuous only for x > 0

Question 26

If f(x) is continuous at x = c, what can be inferred about the graph of f(x) at x = c?

Accepted Answer

The graph has no breaks or discontinuities at x = c

Answer

The graph has a vertical asymptote at x = c

Answer

The graph has a sharp corner at x = c

Question 27

Identify the function that is NOT continuous on the entire real number line:

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 28

Which of the following is a fundamental requirement for the continuity of a function at x = a?

Accepted Answer

The function must be defined at x = a

Answer

The function must have a finite value at x = a

Answer

The function must be increasing on an interval containing x = a

Answer

The function must be differentiable at x = a

Question 29

If the limit of f(x) as x approaches 'a' from the left differs from the limit of f(x) as x approaches 'a' from the right, what conclusion can be drawn about the continuity of f(x) at x = a?

Accepted Answer

f(x) is not continuous at x = a

Answer

f(x) is continuous at x = a

Answer

Continuity of f(x) at x = a depends on its differentiability at x = a

Answer

Continuity of f(x) at x = a depends on its finite value at x = a

Question 30

Which theorem establishes the continuity of functions defined by polynomials?

Accepted Answer

Polynomial Continuity Theorem

Answer

Extreme Value Theorem

Answer

Mean Value Theorem

Answer

Intermediate Value Theorem

Question 31

If f(x) is continuous on the closed interval [a, b], what can be concluded about the existence of its maximum and minimum values on that interval?

Accepted Answer

f(x) is guaranteed to attain both a maximum and a minimum value on [a, b]

Answer

The existence of maximum and minimum values cannot be determined from the continuity of f(x)

Answer

f(x) may have a maximum but not a minimum on [a, b]

Question 32

Which of the following is NOT a potential consequence of a function's discontinuity?

Accepted Answer

The function can be differentiable everywhere

Answer

The function may fail to satisfy the Intermediate Value Theorem

Answer

The function may have 'holes' in its graph

Answer

The function may have vertical asymptotes