Higher-dimensional Case
This above definition can be generalized to real-valued functions f defined on subsets of Rn. Let a be an interior point of the domain of f. Then f is called semi-differentiable at the point a if for every direction u ∈ Rn the limit
exists as a real number.
Semi-differentiability is thus weaker than Gâteaux differentiability, for which one takes in the limit above h → 0 without restricting h to only positive values.
(Note that this generalization is not equivalent to the original definition for n = 1 since the concept of one-sided limit points is replaced with the stronger concept of interior points.)
Read more about this topic: Semi-differentiability
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