Real Analysis Learning Quizzes | Master Mathematics MSc

Question 1

A function is continuous at a point if:

Accepted Answer

Its limit exists and equals the function value at that point.

Answer

It is differentiable at the point.

Answer

It is continuous on some open interval containing the point.

Question 2

The derivative of the function f(x) = x^2 is:

Accepted Answer

2x

Answer

x^2

Answer

1

Answer

0

Question 3

The Cauchy-Riemann equations are primarily used for:

Accepted Answer

Determining if a complex function is analytic.

Answer

Finding the critical points of a function.

Answer

Evaluating the integral of a complex function.

Answer

Solving differential equations.

Question 4

Which of the following is NOT a type of convergence for sequences?

Accepted Answer

Uniform convergence

Answer

Absolute convergence

Answer

Conditional convergence

Answer

Cauchy convergence

Question 5

The Fourier series of a function f(x) is:

Accepted Answer

A series of sine and cosine functions that converges to f(x) on a given interval.

Answer

A series of exponential functions that converges to f(x) on a given interval.

Answer

A series of polynomial functions that converges to f(x) on a given interval.

Answer

A series of rational functions that converges to f(x) on a given interval.

Question 6

The Lebesgue measure is a measure defined on:

Accepted Answer

The real line.

Answer

The plane.

Answer

The Cantor set.

Answer

All sets.

Question 7

Consider the function f(x) = |x| + 1. Determine the limit of f(x) as x approaches 0. Provide a brief explanation of your reasoning.

Accepted Answer

lim_{x→0} f(x) = 1

Answer

lim_{x→0} f(x) = 0

Answer

lim_{x→0} f(x) does not exist

Answer

lim_{x→0} f(x) = ∞

Question 8

Which condition is NOT necessary for the existence of a limit at a point?

Accepted Answer

The function must be differentiable at that point.

Answer

The function must be continuous at that point.

Answer

The left and right-hand limits must exist and be equal.

Answer

The function must be defined at that point.

Question 9

According to the Intermediate Value Theorem, if a function is continuous on a closed interval and takes on values with opposite signs at the endpoints, then:

Accepted Answer

It must have at least one zero within the interval.

Answer

It cannot have any zeros within the interval.

Answer

It may or may not have a zero within the interval.

Answer

It must have an infinite number of zeros within the interval.

Question 10

Calculate the definite integral of the function f(x) = sin(x) over the interval [0, π/2].

Accepted Answer

1

Answer

0

Answer

2

Answer

-1

Question 11

Find the supremum of the set S = {x ∈ ℝ | 0 ≤ x ≤ 1}.

Accepted Answer

1

Answer

0

Answer

2

Answer

1/2

Question 12

Which of the following sets is compact in the Euclidean topology?

Accepted Answer

The set of all real numbers in the interval (0, 1).

Answer

The set of all rational numbers in the interval [0, 1].

Answer

The set of all natural numbers.

Answer

The set of all positive integers.

Question 13

In uniform continuity, what additional condition must a function satisfy on an interval, beyond being continuous?

Accepted Answer

For any positive number $epsilon$, there exists a positive number $delta$ such that for all $x$ and $y$ in the interval, if $|x - y| < delta$, then $|f(x) - f(y)| < epsilon$.

Answer

Its graph is bounded.

Answer

Its derivative is continuous on the interval.

Answer

It is differentiable at every point in the interval.

Question 14

Find the limit of (x^2 - 1) / (x - 1) as x approaches 1:

Accepted Answer

1

Answer

0

Answer

2

Answer

Undefined

Question 15

Calculate the derivative of f(x) = e^(2x):

Accepted Answer

2e^(2x)

Answer

e^x

Answer

e^(2x) / 2

Answer

-e^(2x)

Question 16

Find the integral of sin(x) from 0 to π/2:

Accepted Answer

1

Answer

0

Answer

π/2

Answer

-1

Question 17

State the Cauchy-Schwarz inequality:

Accepted Answer

|⟨x, y⟩| ≤ ||x|| ||y||

Answer

⟨x, y⟩ ≥ ||x|| ||y||

Answer

||x + y|| ≤ ||x|| + ||y||

Answer

||x - y|| ≥ ||x|| - ||y||

Question 18

Which of the following statements best describes the limit of a sequence?

Accepted Answer

The value that the sequence approaches as the index approaches infinity.

Answer

The average of all the elements of the sequence.

Answer

The last element of the sequence.

Answer

The first element of the sequence.

Question 19

Which of the following properties does NOT hold for continuous functions?

Accepted Answer

Preserves injectivity

Answer

Preserves limits

Answer

Preserves connectedness

Answer

Preserves boundedness

Question 20

Find the integral of the function f(x) = sin(x) from 0 to π/2.

Accepted Answer

1

Answer

-1

Answer

0

Answer

π

Question 21

What is the least upper bound (supremum) of the set {1/2, 1/3, 1/4, ...}?

Accepted Answer

1

Answer

Does not exist

Answer

1/2

Answer

0

Question 22

Which of the following sequences is a Cauchy sequence?

Accepted Answer

The sequence {n/(n+1)}

Answer

The sequence {n^2}

Answer

The sequence {(-1)^n}

Answer

The sequence {e^n}

Question 23

What is the total variation of the function f(x) = |x - 1| on the interval [0, 2]?

Accepted Answer

3

Answer

1

Answer

2

Answer

4