Search results
Results from the WOW.Com Content Network
Absolute continuity of measures is reflexive and transitive, but is not antisymmetric, so it is a preorder rather than a partial order. Instead, if μ ≪ ν {\displaystyle \mu \ll \nu } and ν ≪ μ , {\displaystyle \nu \ll \mu ,} the measures μ {\displaystyle \mu } and ν {\displaystyle \nu } are said to be equivalent .
Simple examples of functions that are smooth but not analytic at any point can be made by means of Fourier series; another example is the Fabius function. Although it might seem that such functions are the exception rather than the rule, it turns out that the analytic functions are scattered very thinly among the smooth ones; more rigorously ...
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 ...
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 ...
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
Analogous results for better behaved classes of continuous functions do exist, for example the Lipschitz functions, whose set of non-differentiability points must be a Lebesgue null set (Rademacher's theorem). When we try to draw a general continuous function, we usually draw the graph of a function which is Lipschitz or otherwise well-behaved.
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 ...
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 ...