Weierstrass function

In mathematics called the Weierstrass function is a pathological example of a real-valued function of the real number line. This function has the property that it is continuous everywhere but nowhere differentiable. It is named after its discoverer Karl Weierstrass. Historically, their importance lies in the fact that it is the first satisfactory example of a nowhere differentiable function. However, Weierstrass was not the first to construct such a function. Already more than 30 years earlier, Bernard Bolzano has a function specified, the Bolzanofunktion, but is nowhere differentiable continuous everywhere. However, his proof is incomplete and the construction was not known to a wider professional public. The surprising constructability of such a function changed the common opinion that every continuous function is down to a lot of isolated points, differentiable. The surprise of the former professional community expresses itself, among other things from that at the beginning of the review of the Weierstrass work almost exclusively from the Weierstrass monster is mentioned. (see the history of this function also )

At that time it was assumed intuitively that a continuous function has a derivative, or that the set of points where it is not differentiable, is "small" in some sense. Earlier mathematicians, including Carl Friedrich Gauss, have often assumed that this is true, as Weierstrass points out in his work. This stems from the difficulty to draw or display their quantity is not differentiable points is something other than a finite set of points is a continuous function. However, there are classes of continuous functions that behave better, for example, the Lipschitz continuous functions, the amount of non- differentiable points must be a Lebesgue -null set. If you draw a continuous function, then usually gives the graph of a function that is Lipschitz continuous and other benign properties which do not apply to general, continuous functions.

The Weierstrass elliptic function and the Weierstrass sigma, zeta or eta function are sometimes referred to as the Weierstrass function.

• 2.1 Proof of the continuity
• 2.2 Fractal properties

Weierstrass functions

There are several similar definitions of a Weierstrass function, which are all the series with trigonometric functions.

Definition according to Weierstrass

In Weierstrass ' original work, the function was

With odd and defined, and in addition

Must be met. This work, containing this definition and the proof that the function is nowhere differentiable, has been filed with the Royal Academy of Sciences on July 18, 1872.

Definition according to Hardy

Godfrey Harold Hardy showed in 1916 that the function

Under the assumptions, is nowhere differentiable.

Examples

An often used Weierstrass function is

Clearly, the Weierstrass function

Your graph looks "almost" like the ( differentiable ) sine function. You would also be differentiable only if there finally would be added many summands. Through each of the next ( much smaller ) summands of the graph is changed by a very small piece. It is everywhere convergent, steadily, but you can not construct tangents to it, so it is not differentiable.

Properties

Proof of the continuity

For all and is. In addition, the series is convergent for. Then it follows from the Weierstrass majorant criterion, the (re) uniform convergence of. Because each summand is a continuous function and the limit function of a uniformly convergent sequence of continuous functions must be continuous again, it follows the alleged continuity of.

Fractal properties

The Weierstrass function can perhaps be described as one of the first fractals, although this term was not then used. The function is described in detail at every level, so that if you increase a piece of the curve, it is not progressively approaches a straight line. Regardless of how close one is between two points, the function is monotonic for any even the smallest interval. In his book, The geometry of fractal sets observed Kenneth Falconer, that the Hausdorff dimension of the classical Weierstrass function upward through is limited, and the constants in the above construction, and it is generally assumed that the Hausdorff dimension exactly this value, but this has not been proved yet. The term Weierstrass function is used in the real analysis often used to denote any function with similar features and a similar construction as Weierstrass ' original example. For example, the cosine function in the infinite series by a piecewise linear "zigzag " function to be replaced.

Tightness nowhere differentiable functions

There are an infinite number of continuous nowhere differentiable functions; the Weierstrass function is the classic example where an explicit representation is known. General:

• In topology, it can be shown that the set of nowhere differentiable functions on the interval is dense in the vector space of all continuous real-valued functions on the interval with the topology of uniform convergence.
• In measure theory is also shown that if the space is equipped with the classical Wiener measure, then, the set of functions that are differentiable even at a point in, measure zero. The same is true if one considers only finite subsets of: The nowhere differentiable functions thus form a prevalent subset of.