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for which one takes in the limit above h -> 0 without restricting h to only positive values.

For example, the function is semi-differentiable at , but not Gateaux differentiable there. Indeed, with

(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.)

Properties

* Any convex function on a convex open subset

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