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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 particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
However, this function is not continuously differentiable. A smooth function that is not analytic. The function = {, < is continuous, but not differentiable at x = 0, so it is of class C 0, but not of class C 1.
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 ...
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.
The absolute value is defined and continuous everywhere, and is differentiable everywhere, except for zero. The cubic root is defined and continuous everywhere, and is differentiable everywhere, except for zero. Many common functions are not defined everywhere, but are continuous and differentiable everywhere where they are defined. For example:
These statements are not, in general, true if the function is defined on an open interval (,) (or any set that is not both closed and bounded), as, for example, the continuous function () =, defined on the open interval (0,1), does not attain a maximum, being unbounded above.
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 ...