A function from a connected domain is constant if:

The image is connected but finite

To check continuity at a point, we may verify:

Preimages of basis neighborhoods

If f is continuous and K ⊆ X is compact, then:

Images of compact sets preserve compactness

If f : X → Y is continuous and X is discrete:

Preimages of open sets in Y are open

Only constant functions are continuous

A function f : X → Y is continuous if:

The preimage of every open set in Y is open in X.

The preimage of every closed set in Y is closed in X.

For each x ∈ X, the function preserves neighborhood structure.

Suppose f : X → Y and every preimage of a basis element of Y is open. Then:

Preimages of arbitrary open sets are open

If the codomain Y has the trivial topology, then:

Every function f : X → Y is continuous

Preimages of open sets are automatically open

Only constant functions are continuous

Only injective functions are continuous

If X is finite and has the cofinite topology, every function from X to Y is continuous because:

All subsets of finite sets are cofinite

Preimages of open sets in Y are open in X

The identity map id : X → X is continuous because:

Preimage of any open set is itself

It preserves neighborhood structure

f : X → Y is continuous if:

It preserves limits of sequences (metric spaces)

Preimages of basis elements of Y are open

It maps connected sets to connected sets

It maps compact sets to open sets

Which implies discontinuity?

Preimage of an open set is not open

There exists one open set whose preimage fails to be open

Images of compact sets are not compact

Images of connected sets are not connected

Which imply continuity at a point x₀?

Every neighborhood of f(x₀) has a preimage neighborhood of x₀

Preimages of open sets containing f(x₀) are open

Sequential convergence is preserved in metric spaces

Applying Topological Continuity Quiz

Reviewed by Jede Crisle Cortes Davila

Jede Crisle Cortes Davila, Bachelor of Engineering |

Math Expert

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Jede Crisle D. is a mathematics subject matter expert specializing in Algebra, Geometry, and Calculus. She focuses on developing clear, solution-driven mathematical explanations and has strong experience with LaTeX-based math content. She holds a Bachelor’s degree in Electronics and Communications Engineering.

, Bachelor of Engineering

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