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The absolute value function is continuous (i.e. it has no gaps). It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis. A cusp on the graph of a continuous function. At zero, the function is continuous but not differentiable. If f is differentiable at a point x 0, then f must also be ...
An everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if and only if it has a bounded first derivative; one direction follows from the mean value theorem. In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is ...
If a continuous function on an open interval (,) satisfies the equality () =for all compactly supported smooth functions on (,), then is identically zero. [1] [2]Here "smooth" may be interpreted as "infinitely differentiable", [1] but often is interpreted as "twice continuously differentiable" or "continuously differentiable" or even just "continuous", [2] since these weaker statements may be ...
the function f is n − 1 times continuously differentiable on the closed interval [a, b] and the n th derivative exists on the open interval (a, b), and; there are n intervals given by a 1 < b 1 ≤ a 2 < b 2 ≤ ⋯ ≤ a n < b n in [a, b] such that f (a k) = f (b k) for every k from 1 to n. Then there is a number c in (a, b) such that the n ...
A function of class is a function of smoothness at least k; that is, a function of class is a function that has a k th derivative that is continuous in its domain. A function of class or -function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that ...
The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. [1]
Let f denote a real-valued function defined on a subset I of the real numbers.. If a ∈ I is a limit point of I ∩ [a,∞) and the one-sided limit + ():= + () exists as a real number, then f is called right differentiable at a and the limit ∂ + f(a) is called the right derivative of f at a.
It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1. The mathematical rigor of the delta function was disputed until Laurent Schwartz developed the theory of distributions, where it is defined as a linear form acting on functions.