Continuity in Calculus III: Definition, Types, and Applications

Question 1

Determine the continuity of the function f(x) = { x^2 if x is rational, 0 if x is irrational }.

Accepted Answer

Continuous at every rational number, discontinuous at every irrational number.

Answer

Continuous at every irrational number, discontinuous at every rational number.

Answer

Continuous at both rational and irrational numbers.

Answer

f(x) is differentiable at every rational number.

Question 2

Demonstrate the existence of a zero for the function f(x) = x^3 - 2x^2 + x + 1 within the interval [-1, 2]. Utilize the Intermediate Value Theorem to support your proof.

Accepted Answer

By the Intermediate Value Theorem, since f(-1) = -3 and f(2) = 7, there must exist a c in (-1, 2) such that f(c) = 0.

Answer

Since the function is continuous, it must attain both a maximum and minimum value on the interval [-1, 2]. These values must be either 0 or non-zero.

Answer

Because the function is decreasing on the interval [-1, 2], it cannot have a zero.

Answer

The function's odd symmetry implies a zero at x = 0.

Question 3

Determine the continuity of the function f(x) = |x - 2| at x = 2. Explain your reasoning.

Accepted Answer

Continuous at x = 2 because the left-hand limit, right-hand limit, and function value at x = 2 are equal.

Answer

Continuous at x = 2 because it is defined at that point.

Answer

Not continuous at x = 2 because it has a sharp corner at that point.

Question 4

Which type of discontinuity occurs when the graph can be made continuous by filling in a gap?

Accepted Answer

Removable

Answer

Jump

Answer

Infinite

Question 5

Define continuity of a function at a point.

Accepted Answer

The limit of the function at that point exists and equals the function value at that point.

Answer

The function has a derivative at that point.

Answer

The graph of the function has no breaks or jumps at that point.

Answer

The function is differentiable at that point.

Question 6

Which type of discontinuity occurs when a discontinuity can be removed by redefining the function at a single point?

Accepted Answer

Removable discontinuity

Answer

Jump discontinuity

Answer

Infinite discontinuity

Answer

Essential discontinuity

Question 7

If a function is continuous on a closed interval, what can be concluded about its extreme values?

Accepted Answer

It must have both an absolute maximum and an absolute minimum.

Answer

It may or may not have an absolute maximum and an absolute minimum.

Answer

It must have a relative maximum and a relative minimum.

Answer

It may or may not have a relative maximum and a relative minimum.

Question 8

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

Accepted Answer

|x|

Answer

x^2

Answer

ln(x)

Answer

e^x

Question 9

Which type of discontinuity can result from a removable singularity?

Accepted Answer

Removable discontinuity

Answer

Jump discontinuity

Answer

Infinite discontinuity

Answer

Essential discontinuity

Question 10

Identify the sufficient condition for a function to be continuous on an interval.

Accepted Answer

It has no discontinuities on the interval.

Answer

It is differentiable on the interval.

Answer

It is increasing on the interval.

Answer

It is concave up on the interval.

Question 11

What is a necessary condition for a function to have a removable discontinuity at a point?

Accepted Answer

The function is undefined at that point.

Answer

The function has an infinite limit at that point.

Answer

The function has a jump discontinuity at that point.

Answer

The function has an essential discontinuity at that point.

Question 12

Determine the continuity of the function f(x) = x^2 using the epsilon-delta definition.

Accepted Answer

f(x) is continuous for all real numbers.

Answer

f(x) is discontinuous at x = 0.

Answer

f(x) is continuous only at rational numbers.

Answer

f(x) is continuous only at irrational numbers.

Question 13

Which of the following is NOT a type of discontinuity of a function?

Accepted Answer

Oscillatory discontinuity

Answer

Jump discontinuity

Answer

Removable discontinuity

Answer

Infinite discontinuity

Question 14

Determine the nature of the discontinuity of the function f(x) = |x| at x = 0.

Accepted Answer

Jump discontinuity

Answer

Continuous

Answer

Removable discontinuity

Answer

Infinite discontinuity

Question 15

Which consequence of the Intermediate Value Theorem states that if f(x) is continuous on [a, b] and f(a) < 0 < f(b), then there exists c ∈ (a, b) such that f(c) = 0?

Accepted Answer

If f(x) is continuous on [a, b] and f(a) < 0 < f(b), then there exists c ∈ (a, b) such that f(c) = 0.

Answer

If f(x) is continuous on [a, b] and f(a) ≠ f(b), then there exists c ∈ (a, b) such that f(c) = (f(a) + f(b))/2.

Answer

If f(x) is continuous on [a, b] and f(a) = f(b), then f(x) is constant on [a, b].

Answer

If f(x) is continuous on [a, b] and f(x) > 0 for all x ∈ (a, b), then f'(x) > 0 for all x ∈ (a, b).

Question 16

Identify the function that is NOT continuous at x = 1.

Accepted Answer

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

Answer

f(x) = x^2

Answer

f(x) = |x-1|

Answer

f(x) = e^x

Question 17

Which condition guarantees the continuity of a function f(x) at a point c?

Accepted Answer

The left-hand limit, right-hand limit, and function value at c are all equal.

Answer

The function is differentiable at c.

Answer

The function is continuous on an open interval containing c.

Answer

The function is defined at c.

Question 18

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

Accepted Answer

The function is continuous at c.

Answer

The function has a finite derivative at c.

Answer

The function is increasing at c.

Answer

The function is concave up at c.

Question 19

How can a removable discontinuity of a function f(x) at x = a be removed?

Accepted Answer

Redefine f(a) so that the left-hand and right-hand limits agree.

Answer

Extend the definition of f(x) to all real numbers.

Answer

Take the derivative of f(x).

Answer

Integrate f(x).

Question 20

Which application of the Intermediate Value Theorem is NOT valid?

Accepted Answer

Determining whether a function is invertible.

Answer

Finding the roots of a polynomial equation.

Answer

Approximating the area under a curve.

Answer

Proving the existence of a fixed point.

Question 21

What is the definition of continuity of a function at a point 'a'?

Accepted Answer

The function is defined at 'a' and its limit as 'x' approaches 'a' is equal to 'f(a)'

Answer

The function is differentiable at 'a'

Answer

The function is continuous on an open interval containing 'a'

Answer

The graph of the function has a vertical asymptote at 'a'

Question 22

Which theorem guarantees that if a function is continuous on a closed interval and takes on different values at the endpoints, then it takes on every value between those endpoints?

Accepted Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Answer

Rolle's Theorem

Answer

Cauchy's Mean Value Theorem

Question 23

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

Accepted Answer

The function is continuous on an open interval containing that point and its derivative exists at that point

Answer

The function is continuous at that point

Answer

The function has a finite derivative at that point

Answer

The graph of the function has a slope at that point

Question 24

If a function is continuous on an open interval and its derivative is zero at every point in that interval, what can you conclude about the function?

Accepted Answer

It is constant

Answer

It is linear

Answer

It is quadratic

Answer

It is exponential

Question 25

If a function is not continuous at a point, what could be the cause?

Accepted Answer

Any of the above

Answer

A jump discontinuity

Answer

An essential discontinuity

Answer

A removable discontinuity

Question 26

What is the necessary and sufficient condition for a function f(x) to be continuous at a point 'c'?

Accepted Answer

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

Answer

The derivative of f(x) exists at c.

Answer

f(x) is differentiable at c.

Answer

f(x) is continuous on an interval containing c.

Question 27

Which type of discontinuity occurs when the graph of a function has a hole?

Accepted Answer

Removable discontinuity

Answer

Jump discontinuity

Answer

Infinite discontinuity

Answer

Essential discontinuity

Question 28

Consider the function f(x) = (x - 1)^2. At what point(s) is this function discontinuous?

Accepted Answer

x = 1

Answer

x = 0

Answer

x = -1

Answer

The function is continuous at all real numbers.

Question 29

Determine the negation of the statement: 'If a function is continuous at a point c, then it is differentiable at c.'

Accepted Answer

If a function is continuous at a point c, then it is not differentiable at c.

Answer

If a function is not continuous at a point c, then it is differentiable at c.

Answer

If a function is not differentiable at a point c, then it is continuous at c.

Answer

If a function is differentiable at a point c, then it is not continuous at c.

Question 30

Consider the function f(x) = |x - 2|. Identify the type of discontinuity at x = 2.

Accepted Answer

Jump discontinuity

Answer

Removable discontinuity

Answer

Infinite discontinuity

Answer

The function is continuous at x = 2.

Question 31

Which of the following types of functions may not be continuous on its domain?

Accepted Answer

Rational functions

Answer

Polynomial functions

Answer

Trigonometric functions

Answer

Exponential functions

Question 32

Consider the function f(x) = 1/(x^2 - 4). Identify the type of discontinuity at x = 2.

Accepted Answer

Essential discontinuity

Answer

Removable discontinuity

Answer

Jump discontinuity

Answer

Infinite discontinuity

Question 33

How does the continuity of function $f(x)$ on a closed interval $[a, b]$ enable the calculation of its definite integral $∫_{a}^{b} f(x) dx$?

Accepted Answer

Continuity ensures an unbroken graph of $f(x)$ on $[a, b]$, enabling the calculation of the definite integral as the area under the curve.

Answer

Continuity allows for the application of the Fundamental Theorem of Calculus, which requires differentiability.

Answer

Continuity implies boundedness of $f(x)$ on $[a, b]$, ensuring the existence of a finite integral.

Answer

Continuity implies the existence of an antiderivative of $f(x)$ on $(a, b)$, which can be integrated to find the definite integral.

Question 34

Define continuity precisely using the limit definition. Explain how this definition ensures that a function f(x) is continuous at x = c if the limit of f(x) as x approaches c exists and equals f(c).

Accepted Answer

A function f(x) is continuous at x = c if and only if the following three conditions are satisfied:
1. f(c) is defined.
2. lim x->c f(x) exists.
3. lim x->c f(x) = f(c).

Answer

Continuity means a function's graph has no sudden jumps or breaks.

Answer

Continuity applies only to functions that are differentiable.

Question 35

Which conditions must be satisfied for a function f(x) to be continuous at a point x = c?

Accepted Answer

f(c) exists, is finite, and equals lim_(x->c) f(x).

Answer

f(x) is continuous on its entire domain.

Answer

f(x) is differentiable at x = c.

Question 36

Identify the function that is discontinuous at x = 0:

Accepted Answer

f(x) = |x|

Answer

f(x) = sin(x)

Answer

f(x) = x^2

Answer

f(x) = e^x

Question 37

The Intermediate Value Theorem (IVT) states that if a function f(x) is continuous on the closed interval [a, b], then for any value c between f(a) and f(b):

Accepted Answer

There exists a point d in (a, b) such that f(d) = c.

Answer

The function f(x) is increasing on (a, b).

Answer

The function f(x) is differentiable on (a, b).

Question 38

Which application of the IVT can be used to prove that a function has a root?

Accepted Answer

Bisecting an interval where the function has opposite signs.

Answer

Using the Chain Rule to differentiate the function.

Answer

Evaluating the limit of the function at a critical point.

Question 39

If a graph has a vertical asymptote at x = a, what can be concluded about the continuity of the function at that point?

Accepted Answer

The function is discontinuous at x = a.

Answer

The function is differentiable at x = a.

Answer

The function is continuous at x = a.

Answer

The function has a maximum at x = a.

Question 40

Which function is continuous everywhere?

Accepted Answer

f(x) = x^3 + 2x + 1

Answer

f(x) = sin(x) / x

Answer

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

Answer

f(x) = |x - 2|

Question 41

If a function is differentiable at a point, what can be concluded about its continuity at that point?

Accepted Answer

The function is also continuous at that point.

Answer

Only true for odd functions

Answer

The function is not continuous at that point.

Answer

The function may or may not be continuous.

Question 42

Given f(x) is continuous at x = c and g(x) is continuous at x = f(c), what can be concluded about the continuity of (g∘f)(x) at x = c?

Accepted Answer

The composition function (g∘f)(x) is continuous at x = c.

Answer

The composition function may or may not be continuous at x = c.

Answer

The composition function is continuous only for even functions.

Answer

The composition function is not continuous at x = c.

Question 43

Which of the following conditions ensures that a function is continuous at a given point?

Accepted Answer

The limit of the function at the point exists and equals the value of the function at that point.

Answer

The derivative of the function exists at the point, but this is not sufficient for continuity.

Question 44

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

Accepted Answer

f(x) = sin(1/x)

Answer

f(x) = e^x, which is also continuous everywhere.

Answer

f(x) = x^2, which is a continuous function.

Answer

f(x) = cos(x), which is continuous everywhere.

Question 45

Which is an application of the Intermediate Value Theorem?

Accepted Answer

To demonstrate that a function has a zero within a given interval.

Answer

To identify a function's concavity, which involves examining the second derivative.

Answer

To locate a function's critical points, which is the role of the First Derivative Test.

Question 46

Which function is NOT differentiable at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = sin(x), which is differentiable everywhere.

Answer

f(x) = x^3, which is also differentiable everywhere.

Answer

f(x) = e^x, which is differentiable everywhere.

Question 47

Which condition is necessary for a function to be continuous on an open interval?

Accepted Answer

The function must be continuous at every point within the interval.

Answer

The function must be differentiable at every point within the interval, which is a stronger condition than continuity.

Answer

The function must be bounded within the interval, which is not sufficient for continuity.

Question 48

Consider the following functions. Which one is continuous but not differentiable at x = 1?

Accepted Answer

f(x) = |x - 1|

Answer

f(x) = sin(x), which is also continuous and differentiable everywhere.

Answer

f(x) = e^x, which is continuous and differentiable everywhere.

Answer

f(x) = x^2, which is continuous and differentiable everywhere.

Question 49

Which theorem states that a function that is continuous on a closed interval and differentiable on the open interval within it must have both a maximum and minimum value on the closed interval?

Accepted Answer

Extreme Value Theorem

Answer

Rolle's Theorem, which applies to functions that are continuous on a closed interval and differentiable on the open interval within it, but it only ensures the existence of at least one point where the derivative is zero, not necessarily a maximum or minimum.

Answer

Mean Value Theorem, which relates the average rate of change of a function to its value at two points, not to its maximum or minimum.

Answer

Intermediate Value Theorem, which guarantees the existence of a zero within an interval, not necessarily a maximum or minimum.

Question 50

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

Accepted Answer

f(x) = x^2 + e^x

Answer

f(x) = 1 / x, which is not continuous at x = 0.

Answer

f(x) = sin(x) / x, which is not continuous at x = 0.