Gâteaux derivative ($DI(v)[h]={\lim }_{t\to 0}[I(v+th)-I(v)]/t$)

A functional on a Banach space is Gâteaux differentiable at $v$ when the directional limits exist and form a bounded linear functional of $h$. Fréchet differentiability is stronger, requiring a remainder uniformly small compared with the norm of the increment. A quadratic norm bound on the Taylor remainder gives the stronger property directly. Mere existence of unrelated directional limits would not establish linear differentiability.