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Proposition: Modulus of Continuity is Monotonically Increasing

Let \((X,d_x)\) and \((Y,d_y)\) be metric spaces and let \(f:X\to Y\) be a continuous function on $X$.

The modulus of continuity $\omega_f$ is monotonically increasing, i.e. if $0 \le \delta\le \delta'$ holds for some real numbers $\delta,\delta'$, then $0 \le \omega_f(\delta)\le \omega_f(\delta').$

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Proofs: 1

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References

Bibliography

  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983