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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
Some sources [9] [10] state a sufficient condition for the complex differentiability at a point as, in addition to the Cauchy–Riemann equations, the partial derivatives of and be continuous at the point because this continuity condition ensures the existence of the aforementioned linear approximation. Note that it is not a necessary condition ...
2 Continuity and differentiability. 3 Notation. 4 Rules of computation. ... Under mild conditions (for example, if the function is a monotone or a Lipschitz function ...
In that the existence of uniquely characterises the number ′ (), the fundamental increment lemma can be said to characterise the differentiability of single-variable functions. For this reason, a generalisation of the lemma can be used in the definition of differentiability in multivariable calculus .
In mathematics, strict differentiability is a modification of the usual notion of differentiability of functions that is particularly suited to p-adic analysis. In short, the definition is made more restrictive by allowing both points used in the difference quotient to "move".
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (differentiability class) it has over its domain. [ 1 ] A function of class C k {\displaystyle C^{k}} is a function of smoothness at least k ; that is, a function of class C k {\displaystyle C^{k}} is a function that has a k th ...
The theorem cannot be applied to this function because it does not satisfy the condition that the function must be differentiable for every x in the open interval. However, when the differentiability requirement is dropped from Rolle's theorem, f will still have a critical number in the open interval ( a , b ) , but it may not yield a ...
Let f denote a real-valued function defined on a subset I of the real numbers.. If a ∈ I is a limit point of I ∩ [a,∞) and the one-sided limit + ():= + () exists as a real number, then f is called right differentiable at a and the limit ∂ + f(a) is called the right derivative of f at a.