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The Morton- or z-curve is one example for a space filling curve: Given alevel of refinement L, it maps the interval [0, 2**dL) one-to-one to a set ofd-dimensional cubes of edge length 2**-L that form a subdivision of the unitcube. Similar curves have been proposed for triangular and tetrahedral unitdomains. In contrast to the Hilbert curve that is continuous, the Morton-typecurves produce jumps.  We prove that any contiguous subinterval of the curve divides the domain intoa bounded number of face-connected subdomains. For the hypercube case andarbitrary dimension, the subdomains are star-shaped and the bound is indeedtwo. For the simplicial case in dimensions 2 and 3, the bound is proportionalto the depth of refinement L. We supplement the paper with theoretical andcomputational studies on the frequency of jumps for a quantitative assessment.

Bounds on the number of discontinuities of Morton-type space-filling curves