Let (X, d) be a metric space and f: X → &mathbb{R} be a function. If f is uniformly continuous on X, then which of the following statements matches the definition of uniform continuity?

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Mathematics · Metric Spaces

Verified Solution4 OptionsUpdated: October 2026👁️1 View

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Correct answer

For every ε > 0, there exists δ > 0 such that for all x, y ∈ X, d(x, y) < δ implies |f(x) - f(y)| < ε

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Answer 1

f is uniformly continuous if and only if f is continuous on every open subset of X.

Answer 2

For every δ > 0, there exists ε > 0 such that for all x, y ∈ X, d(f(x), f(y)) < ε implies d(x, y) < δ

Answer 3

For every ε > 0, there exists δ > 0 such that for all x, y ∈ X, d(x, y) < δ implies |f(x) - f(y)| < ε

Answer 4

Uniform continuity of f implies that f is bounded on any metric space X

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