Mathematics · Topology
Let (X, d) be a metric space and f: X → R be a function. Which of the following conditions is equivalent to f being continuous at a point x0 ∈ X?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1For every ε > 0, there exists δ > 0 such that d(x, x0) < ε whenever |f(x) - f(x0)| < δ.
- 2There exists ε > 0 such that |f(x) - f(x0)| 0.
- 3f(x) is bounded in a neighborhood of x0.
- 4For every ε > 0, there exists δ > 0 such that |f(x) - f(x0)| < ε whenever d(x, x0) < δ.✓
Correct answer
For every ε > 0, there exists δ > 0 such that |f(x) - f(x0)| < ε whenever d(x, x0) < δ.
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