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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
In mathematics, the Fréchet derivative is a derivative defined on normed spaces.Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations.
Rademacher's theorem can be used to prove that, for any p ≥ 1, the Sobolev space W 1,p (Ω) is preserved under a bi-Lipschitz transformation of the domain, with the chain rule holding in its standard form. [5] With appropriate modification, this also extends to the more general Sobolev spaces W k,p (Ω). [6]
Since the proof for the standard version of Rolle's theorem and the generalization are very similar, we prove the generalization. The idea of the proof is to argue that if f ( a ) = f ( b ) , then f must attain either a maximum or a minimum somewhere between a and b , say at c , and the function must change from increasing to decreasing (or the ...
Continuity and differentiability This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a jump discontinuity ). The absolute value function is continuous but fails to be differentiable at x = 0 since the tangent slopes do not approach the same value from the left as they do ...
The Vitali covering lemma is vital to the proof of this theorem; its role lies in proving the estimate for the Hardy–Littlewood maximal function.. The theorem also holds if balls are replaced, in the definition of the derivative, by families of sets with diameter tending to zero satisfying the Lebesgue's regularity condition, defined above as family of sets with bounded eccentricity.
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 Gateaux differentiable there.
Let be a function in the Lebesgue space ([,]).We say that in ([,]) is a weak derivative of if ′ = ()for all infinitely differentiable functions with () = =.. Generalizing to dimensions, if and are in the space () of locally integrable functions for some open set, and if is a multi-index, we say that is the -weak derivative of if