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Excerpt from On the Subdifferentiability of Functions of a Matrix Spectrum, Vol. 2: Subdifferential Formulas
The notation e l 0 implies that only real (and positive) values of e are to be considered. The superscript h in (1) is used to emphasize that depends only on the analytic (holomorphic) curves in d?' that pass through z. At first glance it may appear that this notion of directional differentiability is not sufficiently robust to capture the full range of first order variational behavior that is possible for nonlipschitzian functions. However, as we hope to demonstrate, it is sufficiently robust for the analysis of the functions p and a. In order to see why this is so, we need to recall certain properties of the puiseux-newton series expansions for the roots of P, which were introduced by way of example in the discussion following Lemma 1 in Part I of this study. Here we provide a little more detail.
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