author introduces to you an incredible function , which is called Weierstrass function .
In the early days of the development of mathematical analysis, due to the limited tools for studying functions, people have always believed that: , the continuous function is in its definition interval, and , except for the "countable undifferentiable point" , is differentiable . In other words, the non-differentiable points of a continuous function are limited at most:
The function above has only one point b that is non-differentiable
However, for the magical Weierstrass function, it completely subverts the world's cognition.
Is there a line in the world that never bends? This issue is taken seriously by many mathematicians. Many early mathematicians, including Gauss , believed that the non-differentiable part of a continuous function is limited. In 1872, the German mathematician Weierstrass used term series to construct a function that is continuous everywhere but not differentiable everywhere:
The expression of this function looks like this, and it must meet many conditions, otherwise the function will not look like the above:
ab that meets the conditions can take any value
See that w(x) is in this cumulative form. Don’t panic, let’s expand it and see what it looks like:
Weierstrass function
From the shape point of view, the Weierstrass function is a continuous function. The graph seems to have thorns. You will not find a smooth place when you zoom in on any part. At the same time, this function has self-similarity: just like the "infinite matryoshka doll" of 's Mandelbrot set .
Weierstrass function has the property that any local amplification of is similar to the overall . This property is called fractal in mathematics.
The magic of the Weierstrass function is that the slope of each point of its does not exist , is continuous but not differentiable everywhere, and it has no concept of "curve":
At this point, Weierstrass terminated the attempts of mathematicians to prove that continuity implies differentiability, and also made mathematicians even more afraid to rely on intuition or geometric thinking.
