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Semi-differentiability is thus weaker than Gateaux differentiability, 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
The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test whether a critical point x {\displaystyle x} is a local maximum, local minimum, or a saddle point, as follows:
In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers . So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x = a {\textstyle x=a} when
For a function f of three or more variables, there is a generalization of the rule shown above. In this context, instead of examining the determinant of the Hessian matrix, one must look at the eigenvalues of the Hessian matrix at the critical point.
Semiderivative or Semi-derivative may refer to: One-sided derivative of semi-differentiable functions Half-derivative , an operator H {\displaystyle H} that when acting twice on a function f {\displaystyle f} gives the derivative of f {\displaystyle f} .
Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set on the real line and a function defined on with real values.
The precise meaning of the variables and depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form , or analytical significance if the differential is regarded as a ...
Rigorously, a subderivative of a convex function : at a point in the open interval is a real number such that () for all .By the converse of the mean value theorem, the set of subderivatives at for a convex function is a nonempty closed interval [,], where and are the one-sided limits = (), = + ().