We present some results involving the Hyers--Ulam stability of some intermediary point arising from the mean value theorems. First, we make some considerations about the stability of Lagrange mean point and we prove a stability result involving Cauchy mean point. Also, we establish some conditions to the stability of the intermediary point defined by some recent theorems due to Wachnichi or Tong. We investigate similar problems to the integral version of these theorems. We complete with Hyers--Ulam stability of the intermediary point arising from second integral mean value theorem. Also, we present a new integral mean theorem and its stability.