The study of permutation polynomials over finite fields has been a hotspot research topic for a long time. In fact, it is equivalent to the study of one-to-one mapping between finite fields. Therefore, it has many important applications in coding theory, cryptography and algebraic curves, etc. Carlitz had a characterization of permutation polynomials. He proved that if f(x)is a polynomial with coefficients over finite field Fq satisfyingf(0)=f(1) and η(f(a)-f(b)=η(a-b)for every a,b∈Fq, where η is the quadratic character of Fq* . Then f(x)=x^pjfor some integer. In this note, we proved that the above result is also true for any multiplicative character of Fq*.