What is nowhere differentiable?
A function f:S⊆R→R f : S ⊆ ℝ → ℝ is said to be nowhere differentiable. if it is not differentiable at any point in the domain S of f . It is easy to produce examples of nowhere differentiable functions.
What is the meaning of being differentiable function in an interval?
Simply put, differentiable means the derivative exists at every point in its domain. Consequently, the only way for the derivative to exist is if the function also exists (i.e., is continuous) on its domain. Thus, a differentiable function is also a continuous function.
How do you check if a function is continuous and differentiable on an interval?
If it’s derivate exists at every point in the interval. Now if the graph of the derivative over the same interval is continuous, ie. if it has no “holes”, then the derivative exists over the interval and thus the function is differentiable over the interval.
Is there a continuous function that is nowhere differentiable?
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
Why function is differentiable on open interval?
A function, f(x), is continuous on an open interval in its domain if the limit of f(x) as x approaches each value in the interval is equal to f(x) at that value. A function is differentiable at a point if the limit of the difference quotient, (f(x + h) – f(x))/h, as h approaches 0 exists.
Why is differentiability important?
In calculus, a differentiable function is a continuous function whose derivative exists at all points on its domain. Differentiability lays the foundational groundwork for important theorems in calculus such as the mean value theorem. …
Can a function be differentiable on a closed interval?
So the answer is yes: You can define the derivative in a way, such that f′ is also defined for the end points of a closed interval. Note that for some theorem like the mean value theorem you only need continuity at the end points of the interval.
Is every continuous function differentiable give example?
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.
Is a function differentiable on a closed interval?
What are some examples of continuous nowhere differentiable functions?
Indeed, many other examples of continuous nowhere differentiable functions can be found in functions whose graphs are fractals. An example of a continuous nowhere differentiable function whose graph is not a fractal is the van der Waerden function. Nowhere differentiability can be generalized.
What is the Takagi–van der Waerden function?
Introduction The Takagi–van der Waerden functions are defined by (1) where , the distance from x to the nearest integer. The first two are shown in Fig. 1.
Is the graph of the Weierstrass function nowhere differentiable?
The above generalized definition leads to the definition of a nowhere differentiable curve: it is a curve such that it admits a nowhere differentiable parametrization. With this definition, we see that the graph of the Weierstrass function, as a curve, is nowhere differentiable.
How do you prove that a series of functions are continuous?
You just need to apply the Weierstrass M test and the fact that if a series of continuous function converges uniformly to a function f, then f is also continuous Thanks for contributing an answer to Mathematics Stack Exchange!