Continuous functions - relation to compact and connected sets, uniform continuity - Question Bank

1. Which of the following functions is uniformly continuous on the interval (0, infinity)?
A) f(x) = x
B) f(x) = x^2
C) f(x) = sin(x)
D) f(x) = e^x
2. If f is continuous on a compact set K, then f(K) is:
A) Always connected.
B) Always compact.
C) Always open.
D) Always a subset of K.
3. A function f is uniformly continuous on a set S if for every epsilon > 0, there exists a delta > 0 such that for all x, y in S, if |x - y| < delta, then |f(x) - f(y)| < epsilon. This delta is:
A) Dependent on x and y.
B) Dependent on epsilon only.
C) Dependent on epsilon and the specific set S.
D) Dependent on the range of f.
4. Let f: R -> R be a continuous function. If f is uniformly continuous, what can be said about the limit of f(x) as x approaches infinity?
A) The limit must exist and be finite.
B) The limit must be infinity.
C) The limit does not necessarily exist, but f must be bounded.
D) The limit must be zero.
5. Which of the following statements is TRUE?
A) Uniform continuity implies continuity.
B) Continuity implies uniform continuity.
C) Uniform continuity and continuity are equivalent.
D) Uniform continuity is only defined for differentiable functions.
6. If f: X -> Y is continuous, and C is a connected subset of X, then f(C) is guaranteed to be:
A) Compact
B) Connected
C) Open
D) A singleton set
7. Consider the function f(x) = x^3 on the interval [-1, 1]. Is f uniformly continuous on [-1, 1]?
A) No, because the function is not linear.
B) Yes, because [-1, 1] is compact.
C) No, because the derivative is not bounded.
D) Yes, but only if the interval is open.
8. The Heine-Cantor theorem is applicable to continuous functions defined on:
A) Open intervals
B) Unbounded sets
C) Compact sets
D) Any subset of R
9. If f is continuous on a connected set X, then f(X) is:
A) Always compact
B) Always connected
C) Always open
D) Always closed
10. Which of the following is an example of a set that is connected but not compact?
A) [0, 1]
B) (0, 1)
C) {0, 1}
D) R
11. If f: [a, b] -> R is continuous, which property is guaranteed for f?
A) f is uniformly continuous on [a, b].
B) f is uniformly continuous on (a, b).
C) f is uniformly continuous on R.
D) f is uniformly continuous only if f is differentiable.
12. The definition of continuity at a point 'c' states that for every epsilon > 0, there exists a delta > 0 such that if |x - c| < delta, then |f(x) - f(c)| < epsilon. In uniform continuity, what is the key difference regarding delta?
A) Delta depends on epsilon only.
B) Delta depends on epsilon and c.
C) Delta depends on epsilon and x.
D) Delta is zero.
13. Let f: X -> Y be a continuous function. If K is a compact subset of X, then f(K) is always:
A) Open
B) Connected
C) Compact
D) Dense
14. What does it mean for a set to be connected in the context of real analysis?
A) The set can be written as the union of two non-empty disjoint open sets.
B) The set cannot be written as the union of two non-empty disjoint open sets.
C) The set contains only one point.
D) The set is bounded.
15. If f is uniformly continuous on an interval I, and g is uniformly continuous on I, then the composition g(f(x)) is:
A) Always uniformly continuous on I.
B) Uniformly continuous on I only if g is linear.
C) Uniformly continuous on I only if f is surjective.
D) Not necessarily uniformly continuous on I.
16. Which statement is FALSE regarding continuous functions and topological properties?
A) A continuous function maps connected sets to connected sets.
B) A continuous function maps compact sets to compact sets.
C) A continuous function maps open sets to open sets.
D) A continuous function maps closed sets to closed sets (on a compact domain).
17. Consider f(x) = x * sin(1/x) for x != 0 and f(0) = 0. Is f uniformly continuous on [0, 1]?
A) No, because of the discontinuity at x=0.
B) Yes, because the function is bounded and the interval is compact.
C) No, because sin(1/x) oscillates infinitely near 0.
D) Yes, because f'(x) exists everywhere.
18. If f is uniformly continuous on S, and g is uniformly continuous on S, then f - g is:
A) Continuous but not necessarily uniformly continuous.
B) Uniformly continuous on S.
C) Uniformly continuous only if S is compact.
D) Not necessarily continuous.
19. The property that a continuous function maps compact sets to compact sets is known as:
A) Connectedness preservation
B) The Intermediate Value Property
C) The image of a compact set is compact
D) The Uniform Continuity Theorem
20. Let f: R -> R be a continuous function. If f is uniformly continuous, which of the following must be true?
A) f is periodic.
B) f is differentiable.
C) f is bounded.
D) f is surjective.
21. Which of the following conditions guarantees that a continuous function f: (a, b) -> R is uniformly continuous?
A) f is monotonic.
B) f is bounded.
C) The limit of f(x) exists as x approaches a and b.
D) f is differentiable.
22. If f is continuous on X, and C is a connected subset of X, then which of the following is true about f(C)?
A) f(C) is always compact.
B) f(C) is always connected.
C) f(C) is always open.
D) f(C) is always closed.
23. Consider the function f(x) = sqrt(x) on the interval [0, 1]. Is f uniformly continuous on [0, 1]?
A) No, because it's not differentiable at 0.
B) Yes, because [0, 1] is compact.
C) No, because the function grows too quickly.
D) Yes, but only if the interval is open.
24. A function f is continuous on a set S. If S is compact, then f(S) is:
A) Always connected
B) Always compact
C) Always open
D) Always a subset of the domain
25. If f is uniformly continuous on R, then for any sequence (x_n) such that x_n -> infinity, the sequence (f(x_n)) must be:
A) Convergent
B) Bounded
C) Divergent
D) Periodic
26. Let f: X -> Y be a continuous function. If X is a connected space, then f(X) is:
A) A compact subset of Y.
B) A connected subset of Y.
C) An open subset of Y.
D) A closed subset of Y.
27. Uniform continuity is a 'stronger' property than continuity. What does this mean in terms of the delta value in the definition?
A) For uniform continuity, delta can depend on x.
B) For uniform continuity, delta depends only on epsilon and is uniform across the domain.
C) For uniform continuity, delta must be smaller than for continuity.
D) For uniform continuity, delta does not exist.
28. Which of the following functions is NOT uniformly continuous on the interval (0, 1)?
A) f(x) = x
B) f(x) = x^2
C) f(x) = 1/x
D) f(x) = sin(x)
29. The Heine-Cantor theorem implies that a continuous function on a compact set is also:
A) Surjective
B) Injective
C) Uniformly continuous
D) Differentiable
30. If f is continuous on [a, b], the Extreme Value Theorem states that f attains:
A) Its maximum value only.
B) Its minimum value only.
C) Both its maximum and minimum values.
D) Neither its maximum nor minimum value.
31. Which property is NOT necessarily preserved by a continuous function?
A) Connectedness
B) Compactness
C) Boundedness of the image if the domain is bounded
D) Continuity of the image
32. Let f: X -> Y be a continuous map. If X is connected, what can be said about Y?
A) Y must be compact.
B) Y must be connected.
C) Y must be Hausdorff.
D) Y must be complete.
33. If f is continuous on X and K is a compact subset of X, then f(K) is guaranteed to be:
A) Connected
B) Compact
C) Open
D) Dense
34. Let f(x) = sin(x) on R. Is f uniformly continuous on R?
A) No, because R is not compact.
B) Yes, because it is bounded.
C) Yes, because it is periodic.
D) No, because the derivative is not bounded.
35. What is the relationship between uniform continuity and continuity?
A) Uniform continuity implies continuity, but not vice versa.
B) Continuity implies uniform continuity, but not vice versa.
C) They are equivalent concepts.
D) They are unrelated concepts.
36. If f is uniformly continuous on a set S, and g is uniformly continuous on S, is the product f*g necessarily uniformly continuous on S?
A) Yes, always.
B) No, not necessarily. It depends on the boundedness of f and g.
C) No, only if f and g are constant.
D) Yes, if S is compact.
37. Consider the function f(x) = x^2 on the interval [0, 1]. Is f uniformly continuous on [0, 1]?
A) No, because it's not linear.
B) Yes, because [0, 1] is compact.
C) No, because f'(x) = 2x is not constant.
D) Yes, but only if the interval is open.
38. Which theorem states that a continuous function on a closed and bounded interval [a, b] is uniformly continuous on [a, b]?
A) Mean Value Theorem
B) Extreme Value Theorem
C) Heine-Cantor Theorem
D) Intermediate Value Theorem
39. Let f: R -> R be a continuous function. If f is uniformly continuous on R, what can be inferred about the behavior of f(x) as |x| -> infinity?
A) f(x) must approach infinity.
B) f(x) must approach negative infinity.
C) f(x) must be bounded.
D) f(x) must be periodic.
40. A function f is uniformly continuous on a set S if for every epsilon > 0, there exists a delta > 0 such that for all x, y in S, if |x - y| < delta, then |f(x) - f(y)| < epsilon. What is the key difference from the definition of continuity?
A) Delta depends on x and y.
B) Delta depends only on epsilon.
C) Delta depends on epsilon and the specific x and y.
D) Delta depends on the supremum of f.
41. If a function f is uniformly continuous on a set S, and g is also uniformly continuous on S, what can be said about f + g?
A) f + g is uniformly continuous on S.
B) f + g is continuous on S but not necessarily uniformly continuous.
C) f + g is uniformly continuous only if f and g are Lipschitz continuous.
D) f + g is uniformly continuous only if S is compact.
42. What condition on the domain is crucial for a continuous function to be uniformly continuous?
A) The domain must be an open interval.
B) The domain must be a connected set.
C) The domain must be compact.
D) The domain must be a singleton set.
43. Consider the function f(x) = 1/x on the domain (0, 1]. Is f uniformly continuous on (0, 1]?
A) Yes, because it is continuous on (0, 1].
B) No, because the function is unbounded.
C) Yes, because the domain is bounded.
D) No, because the function is not continuous at x=0.
44. The Intermediate Value Theorem is a direct consequence of which property of continuous functions?
A) Mapping compact sets to compact sets.
B) Mapping connected sets to connected sets.
C) Uniform continuity.
D) The definition of continuity using epsilon-delta.
45. Let f: X -> Y be a continuous function and let C be a connected subset of X. What can be said about f(C)?
A) f(C) is always compact.
B) f(C) is always connected.
C) f(C) is always open.
D) f(C) is always closed.
46. What property does a continuous function preserve when mapping a connected set to another topological space?
A) Compactness
B) Connectedness
C) Completeness
D) Boundedness
47. A function f: X -> Y is continuous if and only if the preimage of every open set in Y is an open set in X. What does this imply about the preimage of a closed set?
A) The preimage of every closed set in Y is an open set in X.
B) The preimage of every closed set in Y is a closed set in X.
C) The preimage of every closed set in Y is a compact set in X.
D) The preimage of every closed set in Y is a connected set in X.
48. If f: X -> Y is a continuous function and K is a compact subset of X, then which of the following is true about f(K)?
A) f(K) is always compact in Y.
B) f(K) is always connected in Y.
C) f(K) is always closed in Y.
D) f(K) is always open in Y.
49. Let f: [a, b] -> R be a continuous function on the closed interval [a, b]. What property does the image f([a, b]) possess?
A) It is an open interval.
B) It is a closed interval.
C) It is a compact set.
D) It is a countable set.
50. Which of the following statements about continuous functions and compact sets is TRUE?
A) A continuous function maps a compact set to a compact set.
B) A continuous function maps a compact set to a connected set.
C) A continuous function maps a connected set to a compact set.
D) A continuous function maps a connected set to a connected set.
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