We examine the problem of damage spreading in the off-equilibrium mode coupling equations. The study is done for the spherical p-spin model introduced by Crisanti, Horner and Sommers. For p > 2 we show the existence of a temperature transition T₀ well above any relevant thermodynamic transition temperature. Above T₀ the asymptotic damage decays to zero while below T₀ it decays to a finite value independent of the initial damage. This transition is stable in the presence of asymmetry in the interactions. We discuss the physical origin of this peculiar phase transition which occurs as a consequence of the non-linear coupling between the damage and the two-time correlation functions.

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