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We show that any 2-valued C^{1, \alpha} (\alpha \in (0, 1)) function u ={u_{1}, u_{2}} on an open ball B in {\mathbb R}^{n} with values u_{1}, u_{2}\in {\mathbb R}^{k} whose graph, viewed as a varifold with multiplicity 2 atpoints where u_{1} = u_{2} and with multiplicity 1 at points where u_{1}, u_{2}are distinct, is stationary in the cylinder B \times {\mathbb R}^{k} must be aC^{1, 1/2} function, and the set of its branch points, if non-empty, must haveHausdorff dimension (n-2) and locally positive (n-2)-dimensional Hausdorffmeasure. The C^{1, 1/2} regularity is optimal.

A frequency function and singular set bounds for branched minimal immersions