Continuous functions - relation to compact and connected sets, uniform continuity - One Line Questions
1.
Which of the following is an example of a set that is connected but not compact? —
(0, 1)
2.
Let f: X -> Y be a continuous function. If X is a connected space, then f(X) is: —
A connected subset of Y.
3.
Which of the following statements about continuous functions and compact sets is TRUE? —
A continuous function maps a compact set to a compact set.
4.
Which statement is FALSE regarding continuous functions and topological properties? —
A continuous function maps open sets to open sets.
5.
If f is continuous on a connected set X, then f(X) is: —
Always connected
6.
A function f is continuous on a set S. If S is compact, then f(S) is: —
Always compact
7.
If f is continuous on a compact set K, then f(K) is: —
Always compact.
8.
If f is uniformly continuous on an interval I, and g is uniformly continuous on I, then the composition g(f(x)) is: —
Always uniformly continuous on I.
9.
If f: X -> Y is continuous, and C is a connected subset of X, then f(C) is guaranteed to be: —
Connected
10.
What property does a continuous function preserve when mapping a connected set to another topological space? —
Connectedness
11.
If f is continuous on X and K is a compact subset of X, then f(K) is guaranteed to be: —
Compact
12.
Which property is NOT necessarily preserved by a continuous function? —
Compactness
13.
The property that a continuous function maps compact sets to compact sets is known as: —
The image of a compact set is compact
14.
If f is uniformly continuous on S, and g is uniformly continuous on S, then f - g is: —
Uniformly continuous on S.
15.
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: —
Bounded
16.
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? —
Delta depends on epsilon only.
17.
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? —
Delta depends only on epsilon.
18.
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: —
Dependent on epsilon only.
19.
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? —
f + g is uniformly continuous on S.
20.
Which of the following conditions guarantees that a continuous function f: (a, b) -> R is uniformly continuous? —
The limit of f(x) exists as x approaches a and b.
21.
Let f: R -> R be a continuous function. If f is uniformly continuous, which of the following must be true? —
f is bounded.
22.
If f: [a, b] -> R is continuous, which property is guaranteed for f? —
f is uniformly continuous on [a, b].
23.
If f is continuous on X, and C is a connected subset of X, then which of the following is true about f(C)? —
f(C) is always connected.
24.
Let f: X -> Y be a continuous function and let C be a connected subset of X. What can be said about f(C)? —
f(C) is always connected.
25.
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)? —
f(K) is always compact in Y.
26.
Which of the following functions is NOT uniformly continuous on the interval (0, 1)? —
f(x) = 1/x
27.
Which of the following functions is uniformly continuous on the interval (0, infinity)? —
f(x) = sin(x)
28.
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? —
f(x) must be bounded.
29.
Uniform continuity is a 'stronger' property than continuity. What does this mean in terms of the delta value in the definition? —
For uniform continuity, delta depends only on epsilon and is uniform across the domain.
30.
Let f: [a, b] -> R be a continuous function on the closed interval [a, b]. What property does the image f([a, b]) possess? —
It is a compact set.
31.
If f is continuous on [a, b], the Extreme Value Theorem states that f attains: —
Both its maximum and minimum values.
32.
The Intermediate Value Theorem is a direct consequence of which property of continuous functions? —
Mapping connected sets to connected sets.
33.
Which theorem states that a continuous function on a closed and bounded interval [a, b] is uniformly continuous on [a, b]? —
Heine-Cantor Theorem
34.
Consider the function f(x) = sqrt(x) on the interval [0, 1]. Is f uniformly continuous on [0, 1]? —
Yes, because [0, 1] is compact.
35.
Consider the function f(x) = x^2 on the interval [0, 1]. Is f uniformly continuous on [0, 1]? —
Yes, because [0, 1] is compact.
36.
Consider f(x) = x * sin(1/x) for x != 0 and f(0) = 0. Is f uniformly continuous on [0, 1]? —
Yes, because the function is bounded and the interval is compact.
37.
Let f(x) = sin(x) on R. Is f uniformly continuous on R? —
Yes, because it is periodic.
38.
Consider the function f(x) = x^3 on the interval [-1, 1]. Is f uniformly continuous on [-1, 1]? —
Yes, because [-1, 1] is compact.
39.
Let f: X -> Y be a continuous function. If K is a compact subset of X, then f(K) is always: —
Compact
40.
The Heine-Cantor theorem is applicable to continuous functions defined on: —
Compact sets
41.
The Heine-Cantor theorem implies that a continuous function on a compact set is also: —
Uniformly continuous
42.
What condition on the domain is crucial for a continuous function to be uniformly continuous? —
The domain must be compact.
43.
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? —
The limit does not necessarily exist, but f must be bounded.
44.
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? —
The preimage of every closed set in Y is a closed set in X.
45.
What does it mean for a set to be connected in the context of real analysis? —
The set cannot be written as the union of two non-empty disjoint open sets.
46.
What is the relationship between uniform continuity and continuity? —
Uniform continuity implies continuity, but not vice versa.
47.
Which of the following statements is TRUE? —
Uniform continuity implies continuity.
48.
Let f: X -> Y be a continuous map. If X is connected, what can be said about Y? —
Y must be connected.
49.
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? —
No, not necessarily. It depends on the boundedness of f and g.
50.
Consider the function f(x) = 1/x on the domain (0, 1]. Is f uniformly continuous on (0, 1]? —
No, because the function is unbounded.