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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 ...
Moreover, the fact that the set of non-differentiability points for a monotone function is measure-zero implies that the rapid oscillations of Weierstrass' function are necessary to ensure that it is nowhere-differentiable. The Weierstrass function was one of the first fractals studied, although this term was not used until much later. The ...
Also, symmetric differentiability implies symmetric continuity, but the converse is not true just like usual continuity does not imply differentiability. The set of the symmetrically continuous functions, with the usual scalar multiplication can be easily shown to have the structure of a vector space over R {\displaystyle \mathbb {R ...
The translation in the language of neighborhoods of the (,)-definition of continuity leads to the following definition of the continuity at a point: A function f : X → Y {\displaystyle f:X\to Y} is continuous at a point x ∈ X {\displaystyle x\in X} if and only if for any neighborhood V of f ( x ) {\displaystyle f(x)} in Y , there is a ...
Differentiability is therefore a stronger regularity condition (condition describing the "smoothness" of a function) than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere (see Weierstrass's nowhere differentiable continuous function). It is possible to discuss the existence ...
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
If a real-valued, differentiable function f, defined on an interval I of the real line, has zero derivative everywhere, then it is constant, as an application of the mean value theorem shows. The assumption of differentiability can be weakened to continuity and one-sided differentiability of f. The version for right differentiable functions is ...
In p-adic analysis, the usual definition of derivative is not quite strong enough, and one requires strict differentiability instead. The Gateaux derivative extends the Fréchet derivative to locally convex topological vector spaces. Fréchet differentiability is a strictly stronger condition than Gateaux differentiability, even in finite ...