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Proposition: Modulus of Continuity is Subadditive

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 subadditive, i.e. for all real numbers $\delta\ge 0$, $\delta'\ge 0$, we have $\omega_f(\delta + \delta')\le \omega_f(\delta )+\omega_f(\delta').$

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References

Bibliography

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