An elementary proof of a tangent-cone identity for rank-one planar Lipschitz maps

We give an elementary proof of a tangent-cone identity which might be known to specialists in geometric measure theory. It concerns Lipschitz mappings $g$ from the closed unit ball $B\subset\mathbb{R}^2$ to $\mathbb{R}^2$. Let $x$ be an interior point of differentiability of $g$ at which $Dg(x)$ has rank one. If the upper one-dimensional Hausdorff density of $g(B)$ at $g(x)$ is at most one, then the tangent cone to $g(B)$ at $g(x)$ coincides with the image of $Dg(x)$. This property is used in the proof of rigidity for the four-gradient problem. We provide a detailed, self-contained proof based on differentiability, connectedness and elementary estimates for one-dimensional Hausdorff measure, without using advanced methods from geometric measure theory.