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Proof

(related to Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous)

• Let $[a,b]$ be a closed real interval, $\mathbb R$ be the set of real numbers and \(f:[a,b]\mapsto \mathbb R\) a continuous function.
• We have to show that \(f\) is uniformly continuous.
  • The absolute value of a difference is a distance function which makes the real numbers a metric space.
  • All closed real intervals are compact, so is $[a,b]$.
  • Thus, $f$ is a continuous function mapping a compact domain $[a,b]$ to a metric space $\mathbb R$.
  • Since all continuous functions on compact domains are uniformly continuous , so is $f$.
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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
2. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984