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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
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
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} .
More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map f : U → R m, where U is an open set in R n, is almost everywhere differentiable.
The following version is proven in "Nonlinear programming" (1991). [2] Suppose (,) is a continuous function of two arguments, : where is a compact set.. Under these conditions, Danskin's theorem provides conclusions regarding the convexity and differentiability of the function = (,).
A DAE system of this form is called semi-explicit. [1] Every solution of the second half g of the equation defines a unique direction for x via the first half f of the equations, while the direction for y is arbitrary. But not every point (x,y,t) is a solution of g. The variables in x and the first half f of the equations get the attribute ...
Nearly 70 million Americans rely on Social Security for monthly income. The vast majority, about 65 million, collect Social Security benefits. Another 4.5 million receive Supplemental Security ...
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the L p space ([,]).