Functions of unitaries with Sᵖ-perturbations for non continuously differentiable functions

Consider a function f: T C, n-times differentiable on T and such that its nth derivative f^ (n) is bounded but not necessarily continuous. Let U: R U (H) be a function taking values in the set of unitary operators on some separable Hilbert space H. Let 1<p< and let Sᵖ (H) be the Schatten class of order p on H. If U: t U (t) -U (0) is n-times Sᵖ-differentiable on R, we show that the operator valued function: t R f (U (t) ) - f (U (0) ) Sᵖ (H) is n-times differentiable on R as well. This theorem is optimal and extends several results related to the differentiability of functions of unitaries. The derivatives of are given in terms of multiple operator integrals and a formula and Sᵖ-estimates for the Taylor remainders of are provided.