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In this paper, we study the general question of how characteristics of functional equations influence whether or not they are robust. We isolate examples of properties which are necessary for the functional equations to be robust. On the other hand, we show other properties which are sufficient for robustness. We then study a general class of functional equations, which are of the form $\forall x,y F[f(x-y), f(x+y), f(x),f(y)]=0, where F is an algebraic function. We give conditions on such functional equations that imply robustness. Our results have applications to the area of self-testing/correcting programs. We show that self-testers and self-correctors can be found for many functions satisfying robust functional equations, including algebraic functions of trigonometric functions such as $\tan{x},{1 \over {1+\cot{x}}},$ ${Ax \over {1-Ax}},cosh x.