Abstract
A sufficient condition for a cluster point of a C 1-functio on ℝ2 to be an asymptotic value is given, based on a partitioning into regions of constant valence. We also obtain a sufficient condition for the cluster set of a planar harmonic function to have non-empty interior. An example is given of a planar harmonic function where the image of the critical set is not closed and such that the cluster set has non-empty interior and is a proper subset of the image.
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