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A well-known result due to Fannes is a certain upper bound on the modulus ofcontinuity of the von Neumann entropy with respect to the trace distancebetween density matrices; this distance is the maximum probability ofdistinguishing between the corresponding quantum states. Much more recently,Audenaert obtained an exact expression of this modulus of continuity.  In the present note, Audenaert's result is extended to a broad class ofentropy functions indexed by arbitrary continuous convex functions $f$ in placeof the Shannon--von Neumann function $x\mapsto x\log_2x$. The proof is based onthe Schur majorization.

Modulus of continuity of the quantum $f$-entropy with respect to the trace distance