Note on a function satisfying Palais–Smale condition

We prove that for a function FF which is a finite sum of functions bounded from below, multiplied by positive real numbers, such that any such function has bounded derivative on sets on which it is bounded, we can construct a family of functions which have the same critical set as FF at some level. If FF satisfies the Palais–Smale condition at this level, then any function of the family satisfies the Palais–Smale condition at some level. Conversely, if there exists a function of this family which satisfies the Palais–Smale condition, then FF will satisfy the Palais–Smale condition.