Let n ≥ 3 be an odd integer,d1=3n-1/2-1, d2=3n-2 andube an element of the finite field F3n. This paper shows thatfu(x)=uxd1+xd2is an almost perfect nonlinear (APN) function on F3n if and only if χ(u+1)= χ(u-1)=χ(u), where χ(∙) denotes the quadratic character of F3n. This settles the open problem raised by Ness and Helleseth in IEEE Trans. Inf. Theory 53(7): 2581-2586, 2007, where only the sufficiency part of the result was proved. Furthermore, we investigate the differential spectra offu(x)for elements u satisfying χ(u+1)=χ(u-1) and express them in terms of several quadratic character sums of cubic polynomials.