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section Differentiability classes). If f is differentiable at a point x0, then f must also be continuous at x0. In particular, any differentiable function...

piecewise linear functions (see the figure), piecewise continuous functions, and piecewise differentiable functions. Piecewise functions can be defined using...

a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain. A function of class...

such as imposing complex differentiability in the context of infinite dimensional holomorphy or continuous differentiability in nonlinear analysis. Suppose...

to a Banach space, and Φ a continuous function. If Φ is continuously differentiable we say the system is a differentiable dynamical system. If the manifold...

interval of the real line: Continuously differentiable ⊂ Lipschitz continuous ⊂ α {\displaystyle \alpha } -Hölder continuous, where 0 < α ≤ 1 {\displaystyle...

continuity points), and differentiability of g is interpreted as local absolute continuity (rather than continuous differentiability). Sometimes the given...

of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifically...

{\displaystyle k} times continuously differentiable). If X {\displaystyle X} is an analytic manifold (i.e. infinitely differentiable and charts are expressible...

is differentiable. The AHK differentiability can also be extended to signed measures. Let μ {\displaystyle \mu } be a measure with a continuously differentiable...

geometry. "Differentiability" of a manifold has been given several meanings, including: continuously differentiable, k-times differentiable, smooth (which...

In mathematical analysis, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function...

absolutely continuous ⊆ uniformly continuous = {\displaystyle =} continuous and, for a compact interval, continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely...

a < b: Continuously differentiable ⊂ Lipschitz continuous ⊂ α {\displaystyle \alpha } -Hölder continuous ⊂ uniformly continuous ⊂ continuous, where 0...

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function...

definition, requiring both continuous differentiability and that c be a real number. Since many common functions have continuous derivatives (e.g. polynomials...

complex-differentiable. In particular, holomorphic functions are infinitely complex-differentiable. This equivalence between differentiability and analyticity...

inverse are continuously differentiable. Given two differentiable manifolds M {\displaystyle M} and N {\displaystyle N} , a differentiable map f : M →...

better behaved classes of continuous functions do exist, for example the Lipschitz functions, whose set of non-differentiability points must be a Lebesgue...

continuity and differentiability of the article derivative.) The differentiability of f {\displaystyle f} can be relaxed to one-sided differentiability, a proof...