We prove that if 0 < α < 1 and f is in the Hölder class Λ_α (R), then for arbitrary selfadjoint operators A and B with bounded A - B, the operator f (A) - f (B) is bounded and {norm of matrix} f (A) - f (B) {norm of matrix} ≤ const {norm of matrix} A - B {norm of matrix}^α. We prove a similar result for functions f of the Zygmund class Λ₁ (R): {norm of matrix} f (A + K) - 2 f (A) + f (A - K) {norm of matrix} ≤ const {norm of matrix} K {norm of matrix}, where A and K are selfadjoint operators. Similar results also hold for all Hölder-Zygmund classes Λ_α (R), α > 0. We also study properties of the operators f (A) - f (B) for f ∈ Λ_α (R) and selfadjoint operators A and B such that A - B belongs to the Schatten-von Neumann class S_p. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions. To cite this article: A. Aleksandrov, V. Peller, C. R. Acad. Sci. Paris, Ser. I 347 (2009).