The function: that revolutionizes calculus.

"Plot of Weierstrass function over the interval [−2, 2]. Like some other fractals, the function exhibits self-similarity: every zoom (red circle) is similar to the global plot." (Wikipedia, Weierstrass function)

In the late 19th century (1872) mathematician Karl Weierstrass introduced a formula that is one of the most revolutionary in the world of mathematics. "In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve" (Wikipedia, Weierstrass function) The term fractal curve. 

This means that when we zoom that curve or otherwise use more and more accurate numbers and shorter number distances between the integers we can find more and more roughness in that interesting function. When we make Weierstrass function calculations and curve for the answers we can get quite a smooth curve. But when we start to use decimal numbers that turns the curve rough. The fact is that there is a limitless number of decimal numbers between two characters. 

"Animation based on the increasing of the b value from 0.1 to 5. (Wikipedia, Weierstrass function)

We can continue all decimal numbers as much as we want. So there is an unlimited number of decimal numbers between for example 0,1 and 0,11. That means we can put unlimited numbers between those numbers that seem to be very close in the number line. But we can put 0,1001 or 0,1009 or any number series between 0,1 and 0,11 and the only thing that we must be sure of is that the number series that we give is smaller than 0,11. (0,11>x). 

So Weierstrass formula is the thing that can introduce the situation that the little difference might be a big thing if we look at it in the right scale. The Weiserstrass formula was revolutionized in the 19th. Century. And sometimes people say that this thing is an early fractal. When we research this function we can see that there is a repeating form that is similar to small and large areas or periods. 

"Euler diagram for P, NP, NP-complete, and NP-hard set of problems (excluding the empty language and its complement, which belong to P but are not NP-complete)". (Wikipedia, P versus NP problem). 

"Diagram of complexity classes provided that P ≠ NP. The existence of problems within NP but outside both P and NP-complete, under that assumption, was established by Ladner's theorem. "(Wikipedia, P versus NP problem). 

Weierstrass formula and P=NP. 

Normally P=NP is introduced as the computational or virtual problem. That means the P=NP or P≠NP is used to introduce similarity between P and NP. The non-proven question is: is it always that P=NP or P≠NP? And is that universal? But the P=NP is much more interesting and large-size complexity than other things. 

The Weierstrass formula has a connection with an unproven mathematical millennium problem called P=NP. Can the P-level be the same as the NP level? And can the P level interact with the NP level? That thing is one of the key questions of computing. 

The P=NP means that if we have two identical systems whose size is different when we move something in a smaller system same thing moves in a larger scale system. The P=NP means that when something happens at the P level same thing happens at the NP level. But the major problem is this. In the real world, the P cannot interact directly with the NP. So there forms a medium between those layers or spaces. And the P and NP don't exist in pure forms. The world is full of layers and spaces. Those spaces disturb information that travels between P and NP. 

That thing can cause the idea about the P=NP. The P=NP means that if something changes in the P-level same thing happens in the NP  level. So, P=NP means that we move something in the microcosmos same thing moves in the microcosmos. So the P=NP should be universal. That formula should have an effect between virtual and "real" (material) systems. But the thing is that the energy level that the virtual system can give to the material system is so weak that the virtual system cannot affect the material or physical systems. 

The P=NP problem is the thing that originates in Buddha, Siddharta Gautama. The Buddha introduced the microcosmos is connected with the macrocosmos. So when something happens in the microcosmos that thing reflects immediately into the microcosmos. The fact is that the P and NP levels are not determined and we can say that abstract, virtual mind, or imagination is the P level. 

And physical world is the NP level. If that thing is real, that means the telekinesis is possible. But the fact is that there is needed much higher or stronger force to make the visible interaction between those virtual and physical worlds possible. The P=NP means that if we move something in the macro cosmos we move the same thing in the microcosmos. As I wrote before the P=NP means that all cosmoses or layers. Without depending on their size have a twin in the microcosmos. 

https://www.quantamagazine.org/the-jagged-monstrous-function-that-broke-calculus-20250123/

https://en.wikipedia.org/wiki/NP-intermediate

https://en.wikipedia.org/wiki/P_versus_NP_problem