Mathematics · Real Analysis
Let (fn)n≥1 be a sequence of real-valued functions on ℝ. Which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1If each fn is uniformly continuous and (fn)n≥1 converges to f uniformly, then f is uniformly continuous.✓
- 2If each fn is differentiable and (fn)n≥1 converges to f uniformly, then f is differentiable.
- 3If each fn is bounded and continuous, and (fn)n≥1 converges pointwise to a bounded and continuous function f, then the convergence must be uniform.
- 4If each fn is bounded and (fn)n≥1 converges to f pointwise, then f is bounded.
Correct answer
If each fn is uniformly continuous and (fn)n≥1 converges to f uniformly, then f is uniformly continuous.
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