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A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If x 0 is an interior point in the domain of a function f , then f is said to be differentiable at x 0 if the derivative f ′ ( x 0 ) {\displaystyle f'(x_{0})} exists.
Let be a function in the Lebesgue space ([,]).We say that in ([,]) is a weak derivative of if ′ = ()for all infinitely differentiable functions with () = =.. Generalizing to dimensions, if and are in the space () of locally integrable functions for some open set, and if is a multi-index, we say that is the -weak derivative of if
A function of a real variable is differentiable at a point of its domain, if its domain contains an open interval containing , and the limit = (+) exists. [2] This means that, for every positive real number , there exists a positive real number such that, for every such that | | < and then (+) is defined, and | (+) | <, where the vertical bars denote the absolute value.
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 ...
Constantin Carathéodory's alternative definition of the differentiability of a function can be used to give an elegant proof of the chain rule. [6] Under this definition, a function f is differentiable at a point a if and only if there is a function q, continuous at a and such that f(x) − f(a) = q(x)(x − a).
"Moderate coffee drinking has been related to health benefits," lead study author Lu Qi, M.D., PhD, interim chair of the Department of Epidemiology at Tulane University, told Fox News Digital.
To be a C r-loop, the function γ must be r-times continuously differentiable and satisfy γ (k) (a) = γ (k) (b) for 0 ≤ k ≤ r. The parametric curve is simple if | (,): (,) is injective. It is analytic if each component function of γ is an analytic function, that is, it is of class C ω.
If you see Randy Moss wearing sunglasses on ESPN's "Sunday NFL Countdown," there's a reason for the choice. The Hall of Fame wide receiver-turned-analyst opened Sunday's show to explain that he ...