The authors introduce the concepts of mapping with \(L\)-bounded degree and of mapping with \(L\)-totally bounded degree in an \(L\)-fuzzifying topological vector space in the sense of \textit{U.\,Höhle} [J.~Math.\ Anal.\ Appl.\ 78, 659--673 (1980; Zbl 0462.54002)]; here, \(L\) is a complete distributive lattice with an order reversing involution. Many results and related basic properties are established, mainly the relationship between the continuity and the boundedness of linear operators; that is, a linear operator is continuous if and only if it is bounded. The closed graph theorem for a linear operator is proved as well.