  Various methods can obtain certified estimates for roots of polynomials. Many
applications in science and engineering additionally utilize the value of
functions evaluated at roots. For example, critical values are obtained by
evaluating an objective function at critical points. For analytic evaluation
functions, Newton's method naturally applies to yield certified estimates.
These estimates no longer apply, however, for H\"older continuous functions,
which are a generalization of Lipschitz continuous functions where continuous
derivatives need not exist. This work develops and analyzes an alternative
approach for certified estimates of evaluating locally H\"older continuous
functions at roots of polynomials. An implementation of the method in Maple
demonstrates efficacy and efficiency.
