  The local Lipschitz property is shown for the graph avoiding multiple point
intersection with lines directed in a given cone. The assumption is much
stronger than those of Marstrand's well-known theorem, but the conclusion is
much stronger too. Additionally, a continuous curve with a similar property is
$\sigma$-finite with respect to Hausdorff length and an estimate on the
Hausdorff measure of each "piece" is found.
