Is The Weierstrass Function Differentiable?

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 Weierstrass function periodic?

Here’s a graph of the function. … It is periodic with period 2π.

Can a discontinuous function be differentiable?

If a function is discontinuous, automatically, it’s not differentiable.

How do you know if a function is discontinuous?

If the function factors and the bottom term cancels, the discontinuity at the x-value for which the denominator was zero is removable, so the graph has a hole in it. After canceling, it leaves you with x – 7. Therefore x + 3 = 0 (or x = –3) is a removable discontinuity — the graph has a hole, like you see in Figure a.

How do you know if a function is continuous or differentiable?

If f is differentiable at x=a, then f is continuous at x=a. Equivalently, if f fails to be continuous at x=a, then f will not be differentiable at x=a. A function can be continuous at a point, but not be differentiable there.

Can you integrate the Weierstrass function?

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 weierstrass function a fractal?

Abstract. The Weierstrass-Mandelbrot (W-M) function was first used as an example of a real function which is continuous everywhere but differentiable nowhere. Later, its graph became a common example of a fractal curve.

What means doubly periodic?

A function is said to be doubly periodic if it has two periods and whose ratio is not real. A doubly periodic function that is analytic (except at poles) and that has no singularities other than poles in the finite part of the plane is called an elliptic function (Whittaker and Watson 1990, p.

Is every continuous function is differentiable?

We have the statement which is given to us in the question that: Every continuous function is differentiable. Since, we know that “every differentiable function is always continuous”. … We know that a function is differentiable at c if f′(c)=limh→0f(c+h)−f(c)h exists. Let us check about f(x)=|x| at x = 0.

What functions are not differentiable?

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.

How do you know if a function is differentiable?

A function is said to be differentiable if the derivative of the function exists at all points in its domain. Particularly, if a function f(x) is differentiable at x = a, then f′(a) exists in the domain.

Can a point be continuous but not differentiable?

We see that if a function is differentiable at a point, then it must be continuous at that point. … If is not continuous at , then is not differentiable at . Thus from the theorem above, we see that all differentiable functions on are continuous on .

Is there a function without a derivative?

In the case of functions of one variable it is a function that does not have a finite derivative. For example, the function f(x)=|x| is not differentiable at x=0, though it is differentiable at that point from the left and from the right (i.e. it has finite left and right derivatives at that point).

Why are fractals not differentiable?

As I have understood it, since fractals have infinite iterations, the distance between two points can never decrease, only increase. However, in derivation, the distance between those two points, h, goes towards 0. Hence, this is not possible in fractal curves, even if they are continuous.

Is every continuous function integrable?

Continuous functions are integrable, but continuity is not a necessary condition for integrability. As the following theorem illustrates, functions with jump discontinuities can also be integrable.

Does there exist a function which is continuous everywhere but not differentiable at two points?

Yes, there are some function which are continuous everywhere but not differentiable at exactly two points. … Since we know that modulus functions are continuous at every point, So there sum is also continuous at every point. But it is not differentiable at every point.

Is the Weierstrass function uniformly continuous?

The Weierstrass function f from Equation (1) is everywhere continuous. an cos(bnπx) converges uniformly to f(x) on R.

What does Rolles theorem say?

Rolle’s theorem states that if a function f is continuous on the closed interval and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b.

What makes a function continuous?

Saying a function f is continuous when x=c is the same as saying that the function’s two-side limit at x=c exists and is equal to f(c).

What is the difference between continuous and differentiable?

The difference between the continuous and differentiable function is that the continuous function is a function, in which the curve obtained is a single unbroken curve. It means that the curve is not discontinuous. Whereas, the function is said to be differentiable if the function has a derivative.

How do you tell if a function is continuous from a graph?

A function is continuous when its graph is a single unbroken curve … … that you could draw without lifting your pen from the paper.

How do you know if a function is continuous on an interval?

A function is said to be continuous on an interval when the function is defined at every point on that interval and undergoes no interruptions, jumps, or breaks. If some function f(x) satisfies these criteria from x=a to x=b, for example, we say that f(x) is continuous on the interval .