Property Identification of Continous Images Quiz

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| Questions: 15 | Updated: Dec 15, 2025
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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.

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.

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About This Quiz
Property Identification Of Continous Images Quiz - Quiz

Get ready to explore one of the most important ideas in topology — how connectedness behaves under continuous functions! In this quiz, you’ll deepen your understanding of why a continuous map cannot “break apart” a connected space and how this idea helps classify images of functions. You’ll work with intervals,... see morereal-valued functions, and counterexamples that reveal what continuity can and cannot guarantee. Through each question, you’ll sharpen your reasoning about connected sets, components, and how functions transform them. By the end, you’ll feel more confident identifying when an image must stay connected — and when exceptions occur. Dive in and strengthen your mastery of connectedness through the lens of continuous mappings!
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2) Let X = [0,2] and f(x) = x3 − 3x. Which property of f([0,2]) can be inferred?

Explanation

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

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

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).

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

Explanation

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

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

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)?

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.

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

Explanation

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

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

Explanation

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

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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.

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.

Explanation

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

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

Explanation

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

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

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.

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

Explanation

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

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

Explanation

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

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

Explanation

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

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If f:X→Y is continuous and X is connected, then f(X) cannot be...
Let X = [0,2] and f(x) = x3 − 3x. Which property of f([0,2]) can...
Which of the following is not guaranteed by continuity of f:X→Y on...
A constant function always maps a connected set to a connected set.
Identify all correct statements about continuous images of connected...
Let f:ℝ→ℝ be continuous. Which of the following could be the...
If X is connected and f:X→Y is continuous and injective, then f(X)...
Which property of a connected set X ⊆ ℝ is always reflected in...
Let f:X→Y be continuous. Which of the following properties may...
Continuous images of connected sets in ℝ² must be intervals.
If X = [0,1] and f(x) = cos(x), which property does f([0,1]) have?
Which statements about connectedness and continuous functions are...
If f:X→Y is continuous and f(X) is connected, then X must be...
Let X = [0,5] and f(x) = 2x. Which of the following properties does...
If f:[0,1]→ℝ is defined as f(x)=0, which property does f([0,1])...
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