The authors consider to what extent the results of Nevanlinna theory remain valid if the derivative

${f}^{\text{'}}\left(z\right)$ occurring in many estimates, in particular in the ramification term

${N}_{1}(r,f),$ is replaced by the difference

${\u25b5}_{c}f\left(z\right)=f(z+c)-f\left(z\right)\xb7$ In particular, the authors obtain, for functions of finite order, an analogue of Nevanlinna’s second main theorem in this context. In the counting function

$N(r,1/(f-a\left)\right)$ one can ignore here those

$a$-points of

$f$ which occur in

$c$-separated pairs, that is, points

$z$ for which

$f(z+c)=f\left(z\right)=a,$ provided

${\u25b5}_{c}f\neg \equiv 0\xb7$ A corollary is a version of Picard’s theorem for functions of finite order, where three values occur only in

$c$-separated pairs. The authors also obtain a version of Nevanlinna’s famous five value theorem where points in

$c$-separated pairs are ignored. Applications of the results to difference equations are also given. The results are illustrated by a number of examples. The paper concludes with a discussion of the results and some open questions.