In the text of Gelfand and Fomin on Calculus of Variations, they define the variation of a functional $J$ (if it exists) to be the unique linear functional $\varphi$ satisfying: $$J[y+h] - J[y] = \varphi[h]+\varepsilon\|h\|$$ where $\varepsilon \to 0$ as $\|h\| \to 0$.
Naturally I was curious if the differentiability of $J$ in this sense implied continuity of $J$, as in the case of the usual derivative. The answer seems to be yes if the linear functional $\varphi$ is assumed to be continuous, but I don't see any such assumptions in the text. Is this a standard assumption that is missing from the definition? Or am I wrong to desire such an assumption and we don't need the continuity of $\varphi$ to do meaningful work in this subject?