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In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve. The Weierstrass function has been historically served the role of a pathological function, being the first published ...
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from ...
Continuously differentiable function: differentiable, with continuous derivative. Smooth function: Has derivatives of all orders. Lipschitz function, Holder function: somewhat more than uniformly continuous function. Harmonic function: its value at the center of a ball is equal to the average value on the surface of the ball (mean value property).
A classic example of a pathology is the Weierstrass function, a function that is continuous everywhere but differentiable nowhere. [1] The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions.
Typical examples of functions that are not analytic are The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0. Piecewise defined functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet.
This function is continuous on the closed interval [−r, r] and differentiable in the open interval (−r, r), but not differentiable at the endpoints −r and r. Since f (−r) = f (r), Rolle's theorem applies, and indeed, there is a point where the derivative of f is zero. The theorem applies even when the function cannot be differentiated ...
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 ...
That is, a function is Lipschitz continuous if there is a constant K such that the inequality ((), ()) (,) holds for any ,. [15] The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.
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