Hausdorff dimension of a connected set

A connected subset $C$ of Euclidean space containing distinct points $a,b$ has Hausdorff dimension at least one. The Lipschitz map $x\mapsto |x-a|$ maps $C$ onto a set containing $[0,|b-a|]$, since continuous images of connected spaces are connected. A Lipschitz map cannot increase Hausdorff dimension, whereas an interval has Hausdorff dimension one. Consequently any set of Hausdorff dimension less than one is totally disconnected.