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This above definition can be generalized to real-valued functions f defined on subsets of Rⁿ using a weaker version of the directional derivative. 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, with h ∈ R.

Semi-differentiability is thus weaker than Gateaux differentiability,

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