The target of this paper is to present PARTIAL fuzzy METRIC-preserving functions and characterize the functions $f:[0,1]\to[0,1]$ with this aspect. We give a characterization for PARTIAL fuzzy  METRIC-preserving functions considering the different t-norms. Also, we show that the topology induced by PARTIAL fuzzy METRIC does not preserve under these functions with  an example. Then we  give a characterization of those PARTIAL fuzzy METRIC-preserving functions which preserve completeness and contractivity under some conditions. Finally, we discussed the relation between fuzzy  METRIC preserving and PARTIAL fuzzy preserving functions.

In this paper, we introduce the notion of extended PARTIAL METRIC space and we present some fixed point theorems in generalized PARTIAL METRIC spaces involving linear and nonlinear contractions.