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 Ũ:t ↦ U(t)-U(0) is n-times Sᵖ-differentiable on R, we show that the operator valued function φ : 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 φ are given in terms of multiple operator integrals and a formula and Sᵖ-estimates for the Taylor remainders of φ are provided.