Mathematics · Real Analysis
Let (X, d) be a metric space. A function f: X → ℜ is continuous at x0 if:
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- 1f maps bounded sets to bounded sets
- 2f maps Cauchy sequences to Cauchy sequences
- 3lim f(x) exists as x approaches x0
- 4For every ε > 0, there exists δ > 0 such that d(x, x0) < δ ⇒ |f(x) - f(x0)| < ε✓
Correct answer
For every ε > 0, there exists δ > 0 such that d(x, x0) < δ ⇒ |f(x) - f(x0)| < ε
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