Preimage Behavior in Continuous Functions Quiz

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| Questions: 15 | Updated: Nov 24, 2025
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1) If f is continuous and U ⊆ Y is path-connected, then f^{-1}(U) must be path-connected.

Explanation

Continuity does not preserve path-connectedness in preimages; only images of connected sets remain connected.

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About This Quiz
Preimage Behavior In Continuous Functions Quiz - Quiz

Think you know how preimages behave under continuous functions? This quiz challenges you to apply continuity rules using open sets, closed sets, basis elements, and compact subsets. You’ll examine how preimages relate to density, compactness, and function composition. Through these examples, you'll gain a deeper understanding of how evaluating preimages... see moreprovides a powerful tool for determining continuity in topological spaces. By the end, you’ll confidently analyze continuity using open-set preimage behavior! see less

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2) For a bijection f: X → Y, continuity of f is equivalent to openness of preimages of open sets.

Explanation

Continuity means the preimage of every open set is open; bijectivity is irrelevant.

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3) If f: X → Y and g: Y → Z are functions, (g∘f)^{-1}(W) = f^{-1}(g^{-1}(W)).

Explanation

Preimages respect function composition.

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4) If A ⊆ Y is dense, then f^{-1}(A) is dense in X for every continuous f.

Explanation

Density does not necessarily pull back under continuous maps.

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5) If f is continuous and U ⊆ Y is open, then f(f^{-1}(U)) = U.

Explanation

Only f(f^{-1}(U)) ⊆ U is guaranteed; equality need not hold.

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6) The preimage of a compact set under a continuous function is:

Explanation

Compactness is preserved in images, not preimages.

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7) Which functions are guaranteed to have open preimages of open sets?

Explanation

This is exactly the definition of continuity.

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8) For any function f: X → Y, the preimage of a singleton {y}:

Explanation

A preimage of a point can be any subset of X.

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9) Which can be used to test continuity of f?

Explanation

All three are valid equivalent continuity conditions.

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10) If f is continuous and C ⊆ Y is closed, then:

Explanation

Closed-set preimages remain closed; complements pull back correctly.

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11) Preimages preserve which set operations?

Explanation

Preimages preserve all basic set operations.

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12) If U ⊆ Y is open and f is continuous, then f^{-1}(U) is:

Explanation

This is the definition of continuity.

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13) Let f: X → Y be continuous. Which sets must have closed preimages?

Explanation

Preimages respect intersections and unions of closed sets.

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14) If A ⊆ Y and f^{-1}(A) is empty, which must be true?

Explanation

If no x maps into A, then A contains no point of the image of f.

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15) If the preimage of every basis element of Y is open, then:

Explanation

Open preimages of basis elements are enough to guarantee continuity.

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If f is continuous and U ⊆ Y is path-connected, then f^{-1}(U) must...
For a bijection f: X → Y, continuity of f is equivalent to openness...
If f: X → Y and g: Y → Z are functions, (g∘f)^{-1}(W) =...
If A ⊆ Y is dense, then f^{-1}(A) is dense in X for every continuous...
If f is continuous and U ⊆ Y is open, then f(f^{-1}(U)) = U.
The preimage of a compact set under a continuous function is:
Which functions are guaranteed to have open preimages of open sets?
For any function f: X → Y, the preimage of a singleton {y}:
Which can be used to test continuity of f?
If f is continuous and C ⊆ Y is closed, then:
Preimages preserve which set operations?
If U ⊆ Y is open and f is continuous, then f^{-1}(U) is:
Let f: X → Y be continuous. Which sets must have closed preimages?
If A ⊆ Y and f^{-1}(A) is empty, which must be true?
If the preimage of every basis element of Y is open, then:
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