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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.
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 .
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 ...
A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of electric charge in any volume of space can only change by the amount of electric current flowing into or out of that volume through its boundaries.
Continuity in the Eulerian description is expressed by the spatial and temporal continuity and continuous differentiability of the flow velocity field. All physical quantities are defined this way at each instant of time, in the current configuration, as a function of the vector position x {\displaystyle \mathbf {x} } .
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 ...
The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly. Most functions that occur in practice have derivatives at all points or at almost every point.