In the book "Calculus of Variations" by Gelfand and Fomin, (continuous) linear functionals are defined as follow in the image:
The fact that continuous is in parentheses seems to suggest that they implicitly assume that linear functionals they consider are continuous. This seems also confirmed by the four examples they give just after this definition, which are all four continuous linear functionals.
In the book, they also define the variation or differential of a functional $J$ by the linear functional $\phi$ such that $\Delta J[h] = J[y+h]-J[y] = \phi[h] + \epsilon \Vert h \Vert$, where $\epsilon$ converges to $0$ as $\Vert h \Vert$ converges to $0$. I am not sure if here they assume $\phi$ continuous (in which case the definition of the variation coincides with the Fréchet differential) or if the continuity is not required?
(I saw that a similar question has already been posted on the forum but the answers were not clear)