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More than three decades ago, Boyd and Balakrishnan established a regularityresult for the two-norm of a transfer function at maximizers. Their resultextends easily to the statement that the maximum eigenvalue of a univariatereal analytic Hermitian matrix family is twice continuously differentiable,with Lipschitz second derivative, at all local maximizers, a property that isuseful in several applications that we describe. We also investigate whetherthis smoothness property extends to max functions more generally. We show thatthe pointwise maximum of a finite set of $q$-times continuously differentiableunivariate functions must have zero derivative at a maximizer for $q=1$, butarbitrarily close to the maximizer, the derivative may not be defined, evenwhen $q=3$ and the maximizer is isolated.

On Properties of Univariate Max Functions at Local Maximizers