Conformally invariant metrics and lack of Hölder continuity

The modulus metric between two points in a subdomain of ${\mathbb{R}}^n,n\ge 2,$ is defined in terms of moduli of curve families joining the boundary of the domain with a continuum connecting the two points. This metric is one of the conformally invariant hyperbolic type metrics, which have become a standard tool in geometric function theory. We prove that the modulus metric is not Hölder continuous with respect to the hyperbolic metric.