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Proposition: Modulus of Continuity is Continuous

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 $\omega_f:\mathbb R_+\to\mathbb R_+$ is continuous on the set positive real numbers $\mathbb R_+$, in particular, the limit of $\omega_f$ at $0$ from above is $$\lim_{\delta\searrow 0} \omega_f(\delta)=0.$$

Table of Contents

Proofs: 1

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References

Bibliography

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