Property Identification Of Continous Images Quiz

1) If f:X→Y is continuous and X is connected, then f(X) cannot be expressed as a union of two disjoint non-empty open sets in Y.

True

False

True, because a continuous image of a connected set must itself be connected, and a connected set cannot be expressed as two disjoint nonempty open subsets.

Explanation

True, because a continuous image of a connected set must itself be connected, and a connected set cannot be expressed as two disjoint nonempty open subsets.

2) Let X = [0,2] and f(x) = x³ − 3x. Which property of f([0,2]) can be inferred?

It is disconnected

It contains at least one interval

It is connected

It is compact but disconnected

f is continuous on a connected interval, so its image must be connected, which means it contains all intermediate values.

Explanation

f is continuous on a connected interval, so its image must be connected, which means it contains all intermediate values.

3) Which of the following is not guaranteed by continuity of f:X→Y on connected X?

F(X) is connected

F(X) is path-connected

F(X) is an interval (if X ⊆ ℝ)

F(X) is non-empty if X is non-empty

A connected image need not be path-connected; classic examples include connected sets that are not path-connected (e.g., topologist’s sine curve).

Explanation

A connected image need not be path-connected; classic examples include connected sets that are not path-connected (e.g., topologist’s sine curve).

4) A constant function always maps a connected set to a connected set.

True

False

True, because a constant function maps everything to a single point, which is always connected.

Explanation

True, because a constant function maps everything to a single point, which is always connected.

5) Identify all correct statements about continuous images of connected sets: (Select all that apply)

The image may collapse multiple components into one.

Connectedness of the domain implies connectedness of the image.

Connectedness is always preserved in inverse images.

Continuous maps can break intervals into disconnected pieces.

A continuous function may merge many points or regions into one, but it cannot break a connected set into disconnected pieces. Inverse images do not preserve connectedness.

Explanation

A continuous function may merge many points or regions into one, but it cannot break a connected set into disconnected pieces. Inverse images do not preserve connectedness.

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6) Let f:ℝ→ℝ be continuous. Which of the following could be the image f(ℝ)?

(0,1) ∪ (2,3)

[0,1]

A single point

Both B and C

A continuous image of ℝ must be an interval or a single point. So it can be [0,1] (an interval) or a single point, but never a disconnected union of intervals.

Explanation

A continuous image of ℝ must be an interval or a single point. So it can be [0,1] (an interval) or a single point, but never a disconnected union of intervals.

7) If X is connected and f:X→Y is continuous and injective, then f(X) is connected.

True

False

True, because continuous images of connected sets are connected, and injectivity does not affect the preservation of connectedness.

Explanation

True, because continuous images of connected sets are connected, and injectivity does not affect the preservation of connectedness.

8) Which property of a connected set X ⊆ ℝ is always reflected in f(X) under a continuous map f?

Being closed

Being bounded

Being an interval

Being open

A continuous image of an interval in ℝ is also an interval (by the Intermediate Value Theorem).

Explanation

A continuous image of an interval in ℝ is also an interval (by the Intermediate Value Theorem).

9) Let f:X→Y be continuous. Which of the following properties may fail for f(X) even if X is connected? Select all that apply.

Compactness

Being closed

Being bounded

Being connected

A continuous image of a connected set must stay connected, but it may fail to be closed, bounded, or compact.

Explanation

A continuous image of a connected set must stay connected, but it may fail to be closed, bounded, or compact.

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10) Continuous images of connected sets in ℝ² must be intervals.

True

False

False, because images in ℝ² can be curves, loops, or any connected shape—not necessarily an interval.

Explanation

False, because images in ℝ² can be curves, loops, or any connected shape—not necessarily an interval.

11) If X = [0,1] and f(x) = cos(x), which property does f([0,1]) have?

Connected

Disconnected

Two components

Empty

cos(x) is continuous, so the image of a connected interval is also a connected interval: [cos(1), 1].

Explanation

cos(x) is continuous, so the image of a connected interval is also a connected interval: [cos(1), 1].

12) Which statements about connectedness and continuous functions are correct? Select all that apply.

Continuous images of connected sets are always connected.

A disconnected domain may yield a connected image.

Continuous surjections preserve connectedness in the domain.

Constant functions preserve connectedness.

Continuous maps preserve connectedness; disconnected domains may map to a connected image; and constant functions always produce connected images. Surjectivity does not force the domain to be connected.

Explanation

Continuous maps preserve connectedness; disconnected domains may map to a connected image; and constant functions always produce connected images. Surjectivity does not force the domain to be connected.

Submit

13) If f:X→Y is continuous and f(X) is connected, then X must be connected.

True

False

False, because a disconnected domain can map into a connected image—for example, {-1,1} mapped to {0}.

Explanation

False, because a disconnected domain can map into a connected image—for example, {-1,1} mapped to {0}.

14) Let X = [0,5] and f(x) = 2x. Which of the following properties does f(X) have?

Disconnected

Connected

Empty

Non-continuous

A continuous image of a connected interval remains an interval: f([0,5]) = [0,10], which is connected.

Explanation

A continuous image of a connected interval remains an interval: f([0,5]) = [0,10], which is connected.

15) If f:[0,1]→ℝ is defined as f(x)=0, which property does f([0,1]) exhibit?

Connected

Disconnected

Open interval

Non-empty but disconnected

The image is the single point {0}, which is connected because single points are always connected.

Explanation

The image is the single point {0}, which is connected because single points are always connected.

Property Identification of Continous Images Quiz