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A differentiable function. In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain.In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain.
Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set on the real line and a function defined on with real values.
Download as PDF; Printable version; In other projects ... One can classify functions by their differentiability class. ... ISBN 0-12-050257-7. Bartle, ...
exist and are continuous, where p 1, p 2, …, p n, and p are as above, for all a in the domain, then f is differentiable to order p throughout the domain and has differentiability class C p. If f is of differentiability class C ∞, f has continuous partial derivatives of all order and is called smooth. If f is an analytic function and equals ...
Other functions cannot be differentiated at all, giving rise to the concept of differentiability. A closely related concept to the derivative of a function is its differential. When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x.
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
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One has however the equivalence of strict differentiability on an interval I, and being of differentiability class (i.e. continuously differentiable). In analogy with the Fréchet derivative , the previous definition can be generalized to the case where R is replaced by a Banach space E (such as R n {\displaystyle \mathbb {R} ^{n}} ), and ...