What is meant by nowhere differentiable?

What is meant by 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.

Are there any non-differentiable functions?

Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x. There are however stranger things. The function sin(1/x), for example is singular at x = 0 even though it always lies between -1 and 1.

Where are functions non-differentiable?

A function is non-differentiable where it has a “cusp” or a “corner point”. This occurs at a if f'(x) is defined for all x near a (all x in an open interval containing a ) except at a , but limx→a−f'(x)≠limx→a+f'(x) . (Either because they exist but are unequal or because one or both fail to exist.)

What are the characteristics of a differentiable function?

A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. exists. In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)).

Can the Weierstrass function be integrated?

The antiderivative of the Weierstrass function is fairly smooth, i.e. not too many sharp changes in slope. This just means that the Weierstrass function doesn’t rapidly change values (except in a few places). integrals, unlike derivatives, are highly insensitive to small changes in the function.

Is the Weierstrass function Lipschitz?

Continuous and nowhere Lipschitz An example is given by the Weierstrass function, which is continuous and nowhere differentiable. This can be justified in two ways: A Lipschitz function is differentiable almost everywhere, by Rademacher’s theorem.

How do you know if a function is Nondifferentiable?

A function is not differentiable at a if its graph has a vertical tangent line at a. The tangent line to the curve becomes steeper as x approaches a until it becomes a vertical line. Since the slope of a vertical line is undefined, the function is not differentiable in this case.

What function has no derivative?

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.

What is meant by non differentiable?

Hint: A function f(x) is said to be differentiable, if the derivative of the function exists at every point in its given domain. Then the function is said to be non-differentiable if the derivative does not exist at any one point of its domain.

Are square root functions differentiable?

If x – 1 < 0 (that is x < 1) then √(x – 1) doesn’t exist and hence x is not in the domain of the function. Hence f(x) = √(x – 1) is not differentiable if x < 1. as h approaches zero. In this case when h < 0 the square root doesn’t exist and hence the limit can’t exist.

How do you show that a function is infinitely differentiable?

f is continuously differentiable or C1 if f is differentiable and the function f is continuous. f is infinitely differentiable or C∞ if the nth derivative f(n) is de- fined on all of U for all positive integers n. with the right hand side absolutely convergent. f(x) := { x2 sin(1/x), x = 0, 0, x = 0.

When are functions not differentiable?

When a function is differentiable it is also continuous. But a function can be continuous but not differentiable. For example the absolute value function is actually continuous (though not differentiable) at x=0.

What function is continuous everywhere but not 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.

How to tell if differentiable?

– Differentiable functions are those functions whose derivatives exist. – If a function is differentiable, then it is continuous. – If a function is continuous, then it is not necessarily differentiable. – The graph of a differentiable function does not have breaks, corners, or cusps.

What does it mean for a function to be differentiable?

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.