1. Let jf, g be real-valued functions of class C in R. Functions /, g are called equivalent if there exists a diffeomorphism (of class C°°) r of R such that f°t=g. The main object of this paper is to show under what conditions a function is equivalent to an analytic function (Theorem i). In the case of polynomials, the corresponding result is proved in Thorn ri]. The method of our proof is analogous to that in [1], and our Lemma 3,4 correspond to Theorem R in p.]. Theorem 2 refines Mittag-Lefler's theorem in the real case. The author thanks Mr. Iwasaki for his kind criticisms, and Professor S. Matsuura for his kind encouragement.