Dealing with delay differential equations of the form \[ y^{(n)}(t)=g(t)\,y(t-\tau)+h(t)\text{ \;on \;}[0,b] \] where \(\tau>0\), the notion of Hyers-Ulan stability is first introduced and then investigated via different methods. Popular approachs, such as, iteraction method and fixed point method, are used to obtain the stability results. Moreover, the advantages and disadvantages of each method are discussed. However, the most inovative approach in this paper relies on an open mapping theorem. The latter appears not only in the study of the aforementioned equation, but also in results involving more general linear functional differential equations with constant and multiple delays.