In this paper we introduce and study the concepts of $error$-closed set and $error$-limit ($error$-cluster) points of $L$-nets and $L$-ideals using the notion of almost $N$-compact remoted neighbourhoods in $L$-topological spaces. Then we introduce and study the concept of $error$-continuous mappings. Several characterizations based on $error$-closed sets and the $error$-convergence theory of $L$-nets and $L$-ideals are presented for $error$-continuous mappings.