Continuity in Calculus: Limits and Continuity

Question 1

Which of the following conditions defines the continuity of f(x) at x = a?

Accepted Answer

lim(x->a-) f(x) = lim(x->a+) f(x) = f(a)

Answer

lim(x->a-) f(x) = f(a)

Answer

lim(x->a+) f(x) = f(a)

Question 2

Which of the following statements accurately defines the limit of a function at a point?

Accepted Answer

It represents the function's value as the input approaches that point.

Answer

It represents the function's value at that precise point.

Answer

It exists for all functions without exception.

Answer

It is always zero.

Question 3

Which function is not continuous at x = 0?

Accepted Answer

f(x) = 1/x

Answer

f(x) = x^2

Answer

f(x) = |x|

Answer

f(x) = sin(x)

Question 4

Which of the following accurately defines continuity at a point?

Accepted Answer

The limit of the function as it approaches the point is equal to the function's value there.

Answer

The function has a value at the given point.

Answer

The graph of the function has no breaks at the point.

Answer

The derivative of the function exists at the point.

Question 5

Which statement is not a consequence of the Intermediate Value Theorem?

Accepted Answer

The differentiability of a continuous function is guaranteed.

Answer

A continuous function within an interval assumes all values lying between its minimum and maximum values.

Answer

Given a continuous function and f(a) < 0 < f(b), a point c in the interval exists such that f(c) = 0.

Answer

Continuous functions have no breaks in their graphs.

Question 6

Determine the type of discontinuity of the function f(x) = (x-2)² / (x-2) at x = 2.

Accepted Answer

Removable

Answer

Non-removable

Answer

Infinite

Answer

Jump

Question 7

If the limit of a function as x approaches a from the left is different from the limit as x approaches a from the right, then the function is:

Accepted Answer

discontinuous at a.

Answer

continuous at a.

Answer

differentiable at a.

Answer

undefined at a.

Question 8

Which of the following properties is NOT preserved under composition of continuous functions?

Accepted Answer

Differentiation

Answer

Continuity

Question 9

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

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Question 10

Determine the limit of the function f(x) = (x^2 - 1)/(x-1) as 'x' approaches 1.

Accepted Answer

1

Answer

0

Answer

2

Answer

The limit does not exist.

Question 11

Analyze the function f(x) = x^2 - 4 at x = 2. Which statement accurately describes the function's behavior?

Accepted Answer

The function is continuous at x = 2.

Answer

The function is discontinuous at x = 2 because it is not defined at that point.

Answer

The function is discontinuous at x = 2 because its limit does not exist at that point.

Answer

The function is discontinuous at x = 2 because its graph has a break at that point.

Question 12

Which of the following is a direct consequence of the Intermediate Value Theorem?

Accepted Answer

If a function is continuous on an interval, then it takes on every value between its minimum and maximum values on that interval.

Answer

If a function is continuous on an interval, then it is differentiable on that interval.

Answer

If a function is continuous on an interval, then it is integrable on that interval.

Answer

If a function is continuous on an interval, then its graph is a straight line.

Question 13

What is a necessary condition for a function to be differentiable at a point?

Accepted Answer

The function must be continuous at that point.

Answer

The function must have a derivative at that point.

Answer

The function must have a second derivative at that point.

Answer

The function must be increasing at that point.

Question 14

Provide an example of a function that is continuous but not differentiable?

Accepted Answer

f(x) = |x|

Answer

f(x) = x^3

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 15

Which of the following statements accurately defines continuity at a point a?

Accepted Answer

A function is continuous at a if its limit as x approaches a from the left and from the right both exist and are equal to f(a).

Answer

A function is continuous at a if its limit as x approaches a does not exist.

Answer

A function is continuous at a if its derivative exists at a.

Answer

A function is continuous at a if it has no holes or jumps at a.

Question 16

Which of the following is a consequence of the Intermediate Value Theorem?

Accepted Answer

If $f(a) < 0$ and $f(b) > 0$, then there exists a $c$ between $a$ and $b$ such that $f(c) = 0$.

Answer

If $f(a) < f(b)$, then there exists a $c$ between $a$ and $b$ such that $f(c) = rac{f(a) + f(b)}{2}$.

Answer

If $f$ is continuous on $[a, b]$, then $f$ has a maximum and minimum value on $[a, b]$.

Answer

If $f$ is continuous on $(a, b)$, then $f$ is differentiable on $(a, b)$.

Question 17

Which of the following is NOT a type of discontinuity?

Accepted Answer

Differentiable discontinuity

Answer

Jump discontinuity

Answer

Infinite discontinuity

Answer

Removable discontinuity

Question 18

Determine if the function h(x) = |x - 2| is continuous at x = 2.

Accepted Answer

Continuous at x = 2

Answer

Not continuous at x = 2

Answer

Discontinuous at x = 0

Answer

Continuous for all real x

Question 19

A function is continuous at a point if it has what at that point?

Accepted Answer

A limit that is equal to the function value

Answer

A discontinuity

Question 20

Which property guarantees that there are no breaks or jumps in a continuous function's graph?

Accepted Answer

Continuity

Answer

Integrability

Answer

Differentiability

Question 21

Which function is discontinuous at x = 0?

Accepted Answer

f(x) = 1 / x

Answer

f(x) = e^x

Answer

f(x) = x^2

Answer

f(x) = sin x

Question 22

For a function *f(x)* to be continuous at *x = a*, which of the following conditions must hold true?

Accepted Answer

The limit of *f(x)* as *x* approaches *a* exists, *f(a)* exists, and the limit of *f(x)* as *x* approaches *a* is equal to *f(a)*.

Answer

The function must be defined for all values of *x*.

Answer

The function must be differentiable at *x = a*.

Answer

The limit of *f(x)* as *x* approaches *a* must be infinite.

Question 23

Consider the piecewise function *f(x)* defined as follows:

*f(x) = x² if x < 2*
*f(x) = 4 if x = 2*
*f(x) = x + 2 if x > 2*

At which point is this function discontinuous?

Accepted Answer

x = 2

Answer

x = 1

Answer

The function is continuous everywhere.

Answer

x = 0

Question 24

If *f(x)* is continuous at *x = 3* and *f(3) = 5*, what can we conclude about the limit of *f(x)* as *x* approaches 3?

Accepted Answer

The limit of *f(x)* as *x* approaches 3 is equal to 5.

Answer

The limit of *f(x)* as *x* approaches 3 is equal to 0.

Answer

The limit of *f(x)* as *x* approaches 3 does not exist.

Answer

We cannot conclude anything about the limit based on this information.

Question 25

Which of the following functions is continuous for all real numbers?

Accepted Answer

f(x) = sin(x)

Answer

f(x) = 1/x

Answer

f(x) = |x| / x

Answer

f(x) = tan(x)

Question 26

The Intermediate Value Theorem states that if a function *f(x)* is continuous on a closed interval [a, b], then:

Accepted Answer

For any value *k* between *f(a)* and *f(b)*, there exists a value *c* in the interval [a, b] such that *f(c) = k*.

Answer

The function must be differentiable on the interval [a, b].

Answer

The function must be a polynomial function.

Answer

The function must have a maximum and a minimum value on the interval [a, b].

Question 27

Why is the function *f(x) = 1/x* not continuous at *x = 0*?

Accepted Answer

The function is undefined at *x = 0*.

Answer

The limit of *f(x)* as *x* approaches 0 does not exist.

Answer

The function is not differentiable at *x = 0*.

Answer

The function has a vertical asymptote at *x = 0*.

Question 28

Which of the following functions is an example of a function that is continuous but not differentiable at *x = 0*?

Accepted Answer

f(x) = |x|

Answer

f(x) = sin(x)

Answer

f(x) = x²

Answer

f(x) = x³

Question 29

Which of the following conditions must be met for a function f(x) to be continuous at a point x = a?

Accepted Answer

f(a) exists, lim_(x->a) f(x) exists, and lim_(x->a) f(x) = f(a)

Answer

lim_(x->a) f(x) exists and lim_(x->a) f(x) = f(a).

Answer

f(a) exists and lim_(x->a) f(x) exists.

Question 30

The function f(x) = 1/x is discontinuous at x = 0. What type of discontinuity does it have?

Accepted Answer

Infinite discontinuity

Answer

Jump discontinuity

Answer

Essential discontinuity

Answer

Removable discontinuity

Question 31

Consider the function f(x) = { x^2, if x < 2; 4, if x = 2; x + 2, if x > 2 }.  Is this function continuous at x = 2?

Accepted Answer

No, it has a jump discontinuity.

Answer

No, it has an infinite discontinuity.

Answer

No, it has a removable discontinuity.

Answer

Yes, it is continuous.

Question 32

What is the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2?

Accepted Answer

4

Answer

2

Answer

Does not exist

Answer

0

Question 33

Which of the following functions is continuous on the entire real number line?

Accepted Answer

f(x) = sin(x)

Answer

f(x) = tan(x)

Answer

f(x) = 1/x

Answer

f(x) = |x|

Question 34

If f(x) is continuous at x = c, and g(x) is continuous at x = f(c), then which of the following must also be continuous at x = c?

Accepted Answer

g(f(x))

Answer

f(g(x))

Answer

f(x) / g(x)

Answer

f(x) + g(x)

Question 35

The Intermediate Value Theorem states that if f(x) is continuous on the closed interval [a, b], and k is any number between f(a) and f(b), then there exists a number c in the interval [a, b] such that:

Accepted Answer

f(c) = k

Answer

f(c) = a

Answer

f(c) = 0

Answer

f(c) = b

Question 36

What is the value of 'a' that makes the function f(x) = { x^2, if x < 1; ax + 2, if x ≥ 1 } continuous at x = 1?

Accepted Answer

1

Answer

2

Answer

0

Answer

3

Question 37

If f(x) is continuous on the closed interval [a, b], and f(a) and f(b) have opposite signs, then which of the following is TRUE?

Accepted Answer

There exists at least one root of f(x) in the interval (a, b).

Answer

f(x) must be differentiable on the interval (a, b).

Answer

There exists at least one critical point of f(x) in the interval (a, b).