Continuous Images & Connectedness Quiz - Reasoning & Problem Solving

1) Let X be connected and f:X→Y continuous. Which of the following cannot happen?

F(X) is an interval

F(X) is a single point

F(X) is disconnected

F(X) is a curve without jumps

The continuous image of a connected set must remain connected, so a disconnected image cannot occur.

Explanation

The continuous image of a connected set must remain connected, so a disconnected image cannot occur.

2) Which statements about continuous images of connected sets are always true? Select all that apply.

If X is connected and f is continuous, then f(X) is connected

If f(X) is connected, then X must be connected

If X is connected, then any function from X to ℝ is continuous

If X is connected, images under continuous maps cannot have gaps

Continuous functions preserve connectedness and cannot create gaps. A connected image does not force the domain to be connected, and not all functions on connected sets are continuous.

Explanation

Continuous functions preserve connectedness and cannot create gaps. A connected image does not force the domain to be connected, and not all functions on connected sets are continuous.

Submit

3) Let X = (0,2) and f(x) = floor(x). What can be said about f(X)?

Connected

Disconnected

Empty

Always a single point

The floor function jumps at x = 1. So f((0,2)) = {0,1}, which is disconnected.

Explanation

The floor function jumps at x = 1. So f((0,2)) = {0,1}, which is disconnected.

4) If X is connected and f(X) consists of exactly two points, then:

F is necessarily constant

X must be disconnected

F cannot be continuous

F(X) is interval

A continuous image of a connected set must be connected; a two-point set is disconnected, so continuity must fail.

Explanation

A continuous image of a connected set must be connected; a two-point set is disconnected, so continuity must fail.

5) Suppose f:X→ℝ is continuous and X is connected. Which must be true? Select all that apply.

F(X) is an interval

F is injective

F(X) contains all values between min f(X) and max f(X)

F(X) is either a point or an interval

By the Intermediate Value Theorem, the image of a connected subset of ℝ must be an interval (possibly a single point), containing all intermediate values.

Explanation

By the Intermediate Value Theorem, the image of a connected subset of ℝ must be an interval (possibly a single point), containing all intermediate values.

Submit

6) Let X be connected and f:X→Y continuous. Assume f(X) is disconnected. What conclusion follows?

Y must be disconnected

X must be compact

F must not be continuous

No conclusion can be drawn

A continuous image of a connected set cannot be disconnected; therefore f must not be continuous.

Explanation

A continuous image of a connected set cannot be disconnected; therefore f must not be continuous.

7) Consider X = ℝ \ {0} and f(x) = $\frac {1} {x}$ . What is f(X)?

Connected

Disconnected

Empty

A singleton

f(X) = (-∞,0) ∪ (0,∞), which is disconnected because 0 is missing.

Explanation

f(X) = (-∞,0) ∪ (0,∞), which is disconnected because 0 is missing.

8) A student claims: “If X ⊆ ℝ² is connected and f:X→ℝ is continuous, then f(X) must be a single interval.” This statement is:

Always true

False: image may be disconnected

False: image may be two intervals

False: image may be a single point or a non-degenerate interval

A continuous image of a connected set into ℝ must be an interval or a point. It cannot be disconnected.

Explanation

A continuous image of a connected set into ℝ must be an interval or a point. It cannot be disconnected.

9) Suppose X is connected and f, g are continuous. Which sets are guaranteed connected? Select all that apply.

F(X)

G(f(X))

(f + g)(X)

F(X)g(X)

The continuous image of a connected set is connected. f(X), g(f(X)), and (f+g)(X) are all continuous images of X. The product f(X)g(X) need not be connected.

Explanation

The continuous image of a connected set is connected. f(X), g(f(X)), and (f+g)(X) are all continuous images of X. The product f(X)g(X) need not be connected.

Submit

10) You are told that a function f:X→ℝ has a “broken” graph (a jump). What does this imply?

X is not connected

F(X) must be disconnected

F is not continuous

F is constant

A jump discontinuity means f is not continuous. It does not imply anything about the connectedness of X.

Explanation

A jump discontinuity means f is not continuous. It does not imply anything about the connectedness of X.

11) Let X = [0,1] ∪ [2,3]. Let f(x) = x². Then f(X) is:

Connected

Disconnected

A single interval

Empty

f([0,1]) = [0,1] and f([2,3]) = [4,9]. The image is the union of two separated intervals, hence disconnected.

Explanation

f([0,1]) = [0,1] and f([2,3]) = [4,9]. The image is the union of two separated intervals, hence disconnected.

12) Given a connected space X and continuous f:X→ℝ, which statements are valid? Select all that apply.

If f(X) = {1,2}, the function is discontinuous

If f(X) contains two intervals that don’t touch, f is discontinuous

If f(X) is a single point, the function is constant

If f(X) is unbounded, X must also be unbounded

A continuous image of a connected set cannot have two disconnected intervals. A single-point image implies the function is constant. Unbounded image does not imply the domain is unbounded.

Explanation

A continuous image of a connected set cannot have two disconnected intervals. A single-point image implies the function is constant. Unbounded image does not imply the domain is unbounded.

Submit

13) If X is connected but not path-connected, and f is continuous, what can be said about f(X)?

Must be path-connected

Must be connected

Must be disconnected

Must be an interval

Continuous maps preserve connectedness, but not path-connectedness. So f(X) must be connected, but may or may not be path-connected.

Explanation

Continuous maps preserve connectedness, but not path-connectedness. So f(X) must be connected, but may or may not be path-connected.

14) Let X be connected and f:X→Y continuous. A student claims: “Since X is connected, its image cannot collapse into a point.” Is the student correct?

Yes, always

No, if f is constant the image is a point

Yes, because continuous maps preserve size

Yes, because connected maps cannot shrink sets

False, because a constant function maps every point of X to a single value, producing a single-point (connected) image.

Explanation

False, because a constant function maps every point of X to a single value, producing a single-point (connected) image.

15) Consider a connected set X ⊆ ℝⁿ. For continuous f:X→ℝᵐ, identify all true statements:

F(X) may be a point

F(X) may be an interval (if m = 1)

F(X) cannot have jumps

F(X) must be homeomorphic to X

Continuous images can collapse into a point, or form an interval when mapping into ℝ. Continuous functions have no jumps. But f(X) does not need to be homeomorphic to X.

Explanation

Continuous images can collapse into a point, or form an interval when mapping into ℝ. Continuous functions have no jumps. But f(X) does not need to be homeomorphic to X.

Continuous Images – Reasoning & Problem-Solving Quiz