Master Continuous Functions: A Comprehensive Guide for Real Analysis Students

Question 1

What is the definition of continuity for a function f: X -> Y between metric spaces X and Y?

Accepted Answer

For any x in X and any ε > 0, there exists a δ > 0 such that whenever d(x, x') < δ, then d(f(x), f(x')) < ε.

Answer

For any x in X and any ε > 0, there exists a δ > 0 such that whenever d(f(x), f(x')) < ε, then d(x, x') < δ.

Answer

For any x in X, there exists a δ > 0 such that whenever d(f(x), f(x')) < δ, then d(x, x') < ε.

Answer

For any x in X, there exists an ε > 0 such that whenever d(x, x') < δ, then d(f(x), f(x')) < ε.

Question 2

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

Accepted Answer

Continuous functions are differentiable.

Answer

The composition of continuous functions is continuous.

Answer

The limit of a sequence of continuous functions is continuous if the sequence converges uniformly.

Answer

Continuous functions preserve compactness.

Question 3

Let f: [0, 1] -> R be a function defined by f(x) = x^2 if x is rational and 0 if x is irrational. Determine if f is continuous at x = 0.

Accepted Answer

Yes, f is continuous at x = 0 by the definition of continuity.

Answer

No, f is not continuous at x = 0 because it is not defined at that point.

Answer

Yes, f is continuous at x = 0 by the Heine-Cantor theorem.

Answer

No, f is not continuous at x = 0 because it is not differentiable at that point.

Question 4

Precisely define continuity of a function in the context of metric spaces.

Accepted Answer

Function f: X → Y exhibits continuity at x ∈ X if for any ε > 0, there exists δ > 0 such that d(f(x), f(y)) < ε whenever d(x, y) < δ.

Answer

Function f: X → Y qualifies as continuous if for any open set V ⊆ Y, the inverse image f^(-1)(V) constitutes an open set within X.

Answer

Continuity of function f: X → Y is established if it is differentiable at every point in the domain X.

Answer

Function f: X → Y is continuous if it satisfies Lipschitz continuity.

Question 5

Which of the following precisely defines a continuous function in a metric space (X, d)?

Accepted Answer

For every ε > 0 and every x in X, there exists a δ > 0 such that d(f(x), f(y)) < ε whenever d(x, y) < δ.

Answer

A function that is differentiable at every point in its domain

Answer

A function that is defined on an open interval and has no discontinuities

Answer

A function that preserves the limit of a Cauchy sequence in the domain

Question 6

Which of the following theorems guarantees that there exists a continuous function on a closed and bounded interval [a, b] that takes on all values between its bounds?

Accepted Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Answer

Heine-Cantor Theorem

Answer

Cauchy-Schwarz Inequality

Question 7

Let f(x) be a continuous function on [0, 1]. Which of the following statements is not necessarily true?

Accepted Answer

f(x) is differentiable at every point in [0, 1]

Answer

f(x) is bounded on [0, 1]

Answer

f(x) has a maximum and minimum value on [0, 1]

Answer

f(x) can be approximated by a sequence of polynomials

Question 8

Identify the necessary and sufficient condition for a function to be uniformly continuous on a metric space (X, d).

Accepted Answer

The function is continuous and its domain is compact.

Answer

The function is continuous and its range is bounded.

Answer

The function is differentiable and its derivative is bounded.

Answer

The function is continuous and has a bounded variation.

Question 9

Which statement accurately describes the Heine-Cantor theorem?

Accepted Answer

Every open cover of a compact metric space has a finite subcover.

Answer

It is applicable only to Euclidean spaces.

Answer

It is a necessary and sufficient condition for a metric space to be compact.

Answer

It guarantees the existence of a continuous function on a closed interval that takes on all values between its bounds.

Question 10

Let f(x) be a continuous function on a closed interval [a, b]. Which of the following integrals represents the area under the curve of f(x) on [a, b]?

Accepted Answer

∫[a, b] f(x) dx

Answer

∫[b, a] f(x) dx

Answer

∫[a, b] f'(x) dx

Answer

∫[b, a] f'(x) dx

Question 11

Which statement about continuous functions on a compact metric space is NOT a consequence of the Heine-Cantor theorem?

Accepted Answer

A continuous function on a bounded set is bounded.

Answer

A continuous function on a compact metric space is uniformly continuous.

Answer

A continuous bijective function from a compact metric space to a metric space is a homeomorphism onto its image.

Question 12

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

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = e^x

Question 13

Let (M, d) be a metric space and f : M → R be a function. Which of the following is a necessary and sufficient condition for f to be continuous at a point x ∈ M?

Accepted Answer

For every ε > 0, there exists δ > 0 such that d(f(x), f(y)) < ε whenever d(x, y) < δ.

Answer

f is differentiable at x.

Question 14

Let f : [0, 1] → R be a continuous function. Which of the following is the correct interpretation of the statement 'f is Riemann integrable on [0, 1]'?

Accepted Answer

f can be approximated by a sequence of step functions.

Answer

f is differentiable on (0, 1).

Question 15

Let (X, d) and (Y, ρ) be metric spaces, and let f : X → Y be a continuous function. If {x_n} is a sequence in X that converges to x, prove that the sequence {f(x_n)} in Y converges to f(x).

Accepted Answer

True

Answer

False

Question 16

Which theorem provides a necessary and sufficient condition for a function to be continuous on a compact metric space?

Accepted Answer

Heine-Cantor theorem

Answer

Bolzano-Weierstrass theorem

Answer

Cauchy's criterion

Answer

Rolle's theorem

Question 17

For which interval is the function f(x) = x^2 continuous?

Accepted Answer

(-∞, ∞)

Answer

[-∞, ∞)

Answer

(0, ∞)

Answer

[-1, 1]

Question 18

Which of the following functions is discontinuous at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Answer

f(x) = x^3

Question 19

For a continuous function f: [0, 1] -> [0, 1], which property holds true?

Accepted Answer

f is a homeomorphism.

Answer

f is bijective.

Answer

f is injective.

Answer

f is surjective.

Question 20

Which statement is NOT a consequence of the Heine-Cantor theorem?

Accepted Answer

Continuous functions on compact metric spaces are Riemann integrable.

Answer

Continuous functions on compact metric spaces are uniformly continuous.

Answer

Bounded sequences in compact metric spaces have convergent subsequences.

Answer

Closed and bounded subsets of compact metric spaces are compact.

Question 21

For a continuous function f: [0, 1] -> R, which integral is guaranteed to exist?

Accepted Answer

∫[0, 1] f(x) dx

Answer

∫[0, 2] f(x) dx

Answer

∫[-1, 1] f(x) dx

Answer

∫(0, 1] f(x) dx

Question 22

Which statement about continuous functions is false?

Accepted Answer

The composition of two continuous functions is continuous.

Answer

The sum of two continuous functions is continuous.

Answer

The product of two continuous functions is continuous.

Answer

The inverse of a continuous bijection is continuous.

Question 23

Let X and Y be metric spaces. A function f: X → Y is said to be continuous at a point x ∈ X if and only if:

Accepted Answer

(a) For every ε > 0, there exists a δ > 0 such that if d(x, y) < δ, then d(f(x), f(y)) < ε.

Answer

(b) For every ε > 0, there exists a δ > 0 such that if d(x, y) < ε, then d(f(x), f(y)) < δ.

Question 24

Consider the function f(x) = |x|. Which of the following statements about the continuity of f is correct?

Accepted Answer

(b) f is continuous at x = 0 and at every point in its domain.

Answer

(a) f is continuous at x = 0 but not at any other point.

Answer

(c) f is discontinuous at x = 0 and at every other point in its domain.

Question 25

Which of the following functions is an example of a discontinuous function?

Accepted Answer

(c) f(x) = 1 / x

Answer

(a) f(x) = x^2

Answer

(b) f(x) = sin(x)

Answer

(d) f(x) = e^x

Question 26

Let f: R → R be a function defined by f(x) = { x if x is rational, 0 otherwise }. Is f continuous at x = 0?

Accepted Answer

(c) No, because the limit of f(x) as x approaches 0 does not exist.

Answer

(a) Yes, because f(0) = 0.

Answer

(b) No, because f is not defined at x = 0.

Question 27

Which of the following conditions is sufficient for a function to be continuous on a closed interval [a, b]?

Accepted Answer

(b) f is bounded on [a, b].

Answer

(a) f is differentiable on (a, b).

Answer

(c) f is strictly increasing on [a, b].

Question 28

How do continuous functions aid in modeling the velocity of an object in motion?

Accepted Answer

Scientists and engineers can use continuous functions to determine velocity at any given time.

Answer

Continuous functions enable abrupt velocity changes, shown as jumps in the graph.

Answer

Continuous functions accurately represent rapid accelerations and decelerations as sharp corners on the graph.

Answer

Discontinuities at specific points are acceptable in continuous functions modeling velocity.

Question 29

Explain how the continuity of a function on a closed interval guarantees the existence of a maximum and minimum value, and discuss the implications of this property in optimization problems.

Accepted Answer

Continuity, coupled with the compactness of the interval, guarantees the existence of both a maximum and minimum value.

Answer

Continuity alone guarantees the existence of a maximum and minimum value, regardless of the shape of the function.

Answer

Compactness alone guarantees the existence of a maximum and minimum value, regardless of the continuity of the function.

Question 30

Which of the following is NOT a property of continuous functions on metric spaces?

Accepted Answer

Preservation of one-to-one mappings

Answer

Preservation of limits

Answer

Preservation of connected sets

Answer

Preservation of sequences

Question 31

Let f: X → Y be a function between metric spaces. Which of the following implies that f is continuous at x ∈ X?

Accepted Answer

The limit of f(x) as x approaches x0 is equal to f(x0)

Answer

f(x) is defined

Answer

f is continuous at every point in X

Answer

The limit of f(x) as x approaches x0 exists

Question 32

Let f: X → R be a function on a metric space X. Which of the following is sufficient to ensure that f is continuous?

Accepted Answer

f is uniformly continuous

Answer

f is differentiable

Answer

f is Lipschitz continuous

Answer

f is increasing

Question 33

Let f: R → R be a function defined by f(x) = x^2 if x ≥ 0 and f(0) = 1. Is f continuous at x = 0?

Accepted Answer

Yes, because the limit of f(x) as x approaches 0 is equal to f(0).

Answer

Yes, because f is continuous at every point in its domain.

Answer

No, because f is not differentiable at x = 0.

Answer

No, because the limit of f(x) as x approaches 0 does not exist.

Question 34

Which of the following is a necessary condition for a function to be uniformly continuous on a metric space?

Accepted Answer

The function is continuous.

Answer

The function is bounded.

Answer

The metric space is compact.

Answer

The function is monotone.

Question 35

Let f: R -> R be a function defined by f(x) = |x|. Determine the continuity of f at x = 0.

Accepted Answer

f is continuous at x = 0

Answer

f is discontinuous at x = 0 but can be extended to be continuous at that point

Answer

f is discontinuous at x = 0 from the right

Answer

f is discontinuous at x = 0 from the left

Question 36

For a function f: R -> R, what condition is essential for f to be continuous on the entire real line?

Accepted Answer

f must be continuous at every point in R

Answer

f must be integrable on R

Answer

f must be differentiable at every point in R

Question 37

For a function f(x), if lim[x->a] f(x) exists, what additional condition guarantees the continuity of f(x) at x = a?

Accepted Answer

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

Answer

f(x) is differentiable at x = a

Answer

f(x) is monotonic in an open interval containing a

Answer

f(a) is defined

Question 38

Which theorem states that a continuous function on a closed interval assumes both its maximum and minimum values?

Accepted Answer

Extreme Value Theorem

Answer

Intermediate Value Theorem

Answer

Cauchy-Schwarz Inequality

Answer

Heine-Cantor Theorem

Question 39

Given a continuous function f: [0, 1] -> R, what can be concluded about the set { f(x) : x ∈ [0, 1] }?

Accepted Answer

It is a closed and bounded interval

Answer

It is an open interval

Answer

It is a finite set

Answer

It is an unbounded set

Question 40

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

Accepted Answer

f(x) = |x|/x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Question 41

Let f(x) = {x^2 if x ≠ 0, 1 if x = 0}. Is f(x) continuous at x = 0?

Accepted Answer

No

Answer

Yes, because lim_{x→0} f(x) = 0.

Answer

Yes, because f(0) = 1.

Question 42

If f: [a, b] → R is continuous, then f is:

Accepted Answer

Bounded on [a, b].

Answer

Always differentiable on [a, b].

Answer

Always strictly increasing on [a, b].

Answer

Always injective on [a, b].

Question 43

Which of the following theorems guarantees the existence of both a minimum and maximum value for a continuous function on a closed interval?

Accepted Answer

Extreme Value Theorem

Answer

Intermediate Value Theorem

Answer

Heine-Cantor Theorem

Answer

Mean Value Theorem

Question 44

Given a continuous function f: R → R and a sequence (x_n) converging to a point x, what can we conclude about the sequence (f(x_n))?

Accepted Answer

(f(x_n)) converges to f(x).

Answer

(f(x_n)) oscillates.

Answer

(f(x_n)) diverges.

Answer

(f(x_n)) converges to 0.

Question 45

Consider the function f(x) = 1/x defined on (0, 1). Is f(x) uniformly continuous on (0, 1)?

Accepted Answer

No

Answer

Yes, because f(x) is differentiable on (0, 1).

Answer

Yes, because f(x) is continuous on (0, 1).

Answer

Yes, because f(x) is bounded on (0, 1).

Question 46

Which of the following functions is NOT continuous on its domain?

Accepted Answer

f(x) = |x| / x

Answer

k(x) = e^x

Answer

h(x) = x^2 + 1

Answer

g(x) = sin(x)

Question 47

The Heine-Cantor theorem implies that a continuous function on a compact set is:

Accepted Answer

Uniformly continuous.

Answer

Differentiable.

Answer

Bounded.

Answer

Injective.

Question 48

Let f(x) be a continuous function on [a, b] and let c be a point in (a, b). If f(c) > f(x) for all x in [a, b] except x = c, then c is:

Accepted Answer

A local maximum of f(x).

Answer

A point of inflection of f(x).

Answer

A discontinuity of f(x).

Answer

A local minimum of f(x).

Question 49

If a function f(x) is continuous on [a, b] and differentiable on (a, b), then the Mean Value Theorem states that there exists a point c in (a, b) such that:

Accepted Answer

f'(c) = (f(b) - f(a)) / (b - a).

Answer

f'(c) = 0.

Answer

f(c) = f(a) + f(b).

Answer

f(c) = (f(b) - f(a)) / (b - a).

Question 50

Which of the following statements is **FALSE** about continuous functions?

Accepted Answer

A function is continuous if it has a discontinuity at every point in its domain.

Answer

The composition of two continuous functions is continuous.

Answer

The sum and product of two continuous functions are continuous.

Question 51

The function f(x) = x^2 is continuous at x = 2. What does this mean?

Accepted Answer

The limit of f(x) as x approaches 2 exists and is equal to f(2).

Answer

The function has an essential discontinuity at x = 2.

Answer

The function has a removable discontinuity at x = 2.

Answer

The function has a jump discontinuity at x = 2.

Question 52

Let f(x) = |x| and g(x) = x. What is the continuity of the composite function h(x) = f(g(x))?

Accepted Answer

h(x) is continuous for all real numbers.

Answer

h(x) is discontinuous at x = 0.

Answer

h(x) is discontinuous for all x ≠ 0.

Answer

h(x) is continuous only for positive values of x.

Question 53

Which of the following functions is **NOT** uniformly continuous on the interval (0, 1)?

Accepted Answer

f(x) = 1/x

Answer

f(x) = x + 1

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Question 54

If a function f(x) is continuous on a closed interval [a, b], what is guaranteed about its maximum and minimum values?

Accepted Answer

The function f(x) attains its maximum and minimum values on the interval [a, b].

Answer

The function f(x) attains its maximum value but not its minimum value on the interval [a, b].

Answer

The function f(x) may not attain its maximum and minimum values on the interval [a, b].

Question 55

If a function f(x) is continuous on a closed interval [a, b] and f(a) ≠ f(b), what is guaranteed?

Accepted Answer

For any value k between f(a) and f(b), there exists at least one point c in the interval (a, b) such that f(c) = k.

Answer

The function f(x) is differentiable on the interval [a, b].

Question 56

Consider the function f(x) defined as follows: f(x) = x^2 for x ≤ 1 and f(x) = x for x > 1. Is this function continuous at x = 1?

Accepted Answer

The function is not continuous at x = 1.

Answer

The function is continuous at x = 1, but the limit does not exist.

Answer

The function is discontinuous at x = 1, but the limit exists.

Answer

The function is continuous at x = 1, and the limit exists.