Search results
Results from the WOW.Com Content Network
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 .
A discontinuous function is a function that is not continuous. Until the 19th century, ... is everywhere continuous. However, it is not differentiable at = ...
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 ...
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.
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. In fact, using the Baire category theorem, one can show that continuous functions are generically nowhere differentiable. [2]
The exponential function becomes arbitrarily steep as x → ∞, and therefore is not globally Lipschitz continuous, despite being an analytic function. The function f(x) = x 2 with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x approaches infinity. It is however locally Lipschitz continuous.
Any continuous function on the interval [,] is also uniformly continuous, since [,] is a compact set. If a function is differentiable on an open interval and its derivative is bounded, then the function is uniformly continuous on that interval.
The graph of the Cantor function on the unit interval. 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 ...