Abstract: Equidistance point and difference theory in theory of function approximation are studied.Meanwhile,the relation among difference,difference quotient and derivate is revealed.By drawing Lagrange's and Cauchy's theorem of mean on difference and Taylor's formula into difference function,four theorems,such as Lagrange's theorem of mean on difference,are concluded in simple way.On the basis of these conclusions,the asymptotic property of middle point is studied,a series of new conclusions are drawn and the discussions on the asymptotic property of middle point in differential mid-value are summarized.Some proof methods for theory of function approximation are improved by means of the given lemmas in this paper and some contents of theory of function approximation are simplified.