Continuous but non-derivable functions: mathematical curiosity or reflection of reality?
In mathematics, when thinking about functions, the most common thing is to come up with simple examples: polynomials, rational functions, sinus and cosine and/or exponential and logarithm, among others. All of them have an excellent mathematical behavior. But there are also those who are more erratic... and they serve to describe reality!
Mathematical analysis is a major field of mathematics that studies numbers and their relationships at the base. Among these relationships, functions are of great importance. The numbers are well known, but what is a function? Well, it's just a rule that assigns another number to a number. Although it is a simple concept, they are an essential tool of mathematics and science. The speed of a satellite, the number of people infected by a disease, the fluctuations of the stock exchange, the temperature that we will have tomorrow... In general, any variable with numerical value is represented by a function.
The most common is to call a general function, which assigns a different number to any number. To give you an example, let’s say it is. This function assigns its double to any number: the number, and the number. We can also define square, inverse, sine and/or exponential. These functions are very common and useful. For example, sound and radio waves have a sinuous appearance, and population growth tends to be exponential. These functions have enormous mathematical properties: they are continuous and can be derived as many times as desired.
Representing functions: graphs
A simple way to become aware of these properties is to represent the functions, which is done through the graph (see Figure 1). In the plane and perpendicular axes are drawn, and by setting, points with coordinates are drawn. The result is a curve and each function corresponds to a different curve. as shown in Figure 2, the function corresponds to a straight line, which is a parabola and a hyperbola.
The graph of a function clearly shows some of the properties of the function. A continuous function if the damping curve is not interrupted. The values of these functions change without jumping. For example, the functions of FIG. 2 are continuous. Otherwise, the function is said to be discontinuous (see Figure 3).
Another important property of functions is their derivatization. The function is derivable (or differentiable) if your graph has a smooth appearance, without corners. The values of these functions are not subject to abrupt trend changes. For example, the graphs in FIG. 2 are smooth and correspond to derivable functions, but the graph in FIG. 4 has two corners so that this function is not derivable.
Often, rather than globally, the continuity and derivation of a function is analyzed by points or intervals. In fact, a function may be continuous or derivable at some points and not at others, as shown in Figure 5 .
These two properties are linked by a base-to-base relationship, since a derivable function is always continuous. Of course, a curve that is smooth has no jump, because a jump would break the smoothness! But what about the opposite? Are continuous functions necessarily derivable?
Continuous but non-derivable functions
The answer is no, of course. The function can be continuous without being derivable, because a continuous graph can have corners. an example of this is shown in Figure 4. Moreover, we can easily draw a curve with as many corners as we want, as shown in Figure 6.
We'll try to do a little more. Is it possible for a continuous curve to have corners at each and every point? Intuition, or simply trying it by hand, says no. But that’s a strange question! More than one will think if it makes sense. Well, that was one of the great puzzles of 19th century mathematicians. At that time, the theoretical foundations of mathematics had not yet been fully established, and the concepts had to be carefully defined so as not to create contradictions. As for this question, most of them thought it was impossible to make such a graph. Said in mathematical language, it was not possible to create a continuous function that was not derivable anywhere.
Half-corrupt convictions. According to Karl Weierstrass [W], in the 1860s, Bernhard Riemann proposed an exotic continuous function and said it had no derivatives anywhere. The function looks like this:
Okay, it's more complicated than the first ones. But even the function with strange properties will not be strange? To begin with, the result of this infinite sum is always a finite number, whatever it may be. In fact, the sine of any number is always less than 1, and Leonhard Euler proved in 1735 that it is the result of the sum of the inverse squares. The continuity of the function was also not difficult to prove. As for the drift, however, no one was able to prove what Riemann said. It was a mystery until 1970 [G].
One hundred years, baby! This function is not a midnight cough in the goat. Its name is a non-differentiable function of Riemann, and as you can imagine, it is almost impossible to draw its graph by hand. Thank goodness we have computers! See Figures 7 and 8. I miss him so much. To be perfectly correct, the Riemann function is not derived almost anywhere, since a straight line is created in the middle of the graph, in the coordinate. There the function has a derivative, but nowhere else.
The Riemann function began the career of continuous and non-derivable functions. Many were proposed, but many mathematicians opposed them, saying that they made no sense. Henri Poincaré, for example, said: “The purpose of these functions seems to be to have as little resemblance as possible to functions that serve something” [P]. But more and more examples were emerging, and especially after the development of the Brownian movement, continuous and non-derivable functions were fully accepted. Then, in the 1970s, Benoît Mandelbrot proposed fractals in order to unify all these functions, in the belief that they could help to describe natural events. In fact, in his words, “the clouds are not spheres, the mountains are not cones, the coasts are not circles and the bark of the trees is not smooth. Even lightning does not extend in a straight line.”[M, 1. by the Orr. ]. There is no lack of reason.
The function of Riemann in nature
In reality, what appears in the physical form is a version of Riemann's function, the following
This expression shows complex numbers that are generated with the complex unit. We are not going to talk about them today, but it is clear that this new function has the same mathematical structure as the non-differentiable function of Riemann. Its representation is shown in Figure 11 .
In a surprising way, the professors of the UPV De la Hoz and Vega showed that this function appears in the movement of the rings of smoke [H-V]. We all know how smoke rings expand: when a smoker creates one of these, the smoke maintains a circular appearance while traveling, at least initially. But what if the ring has a triangular or quadrilateral shape? This is not a rare situation, as triangular and square tubes are used in the industry. This experiment was performed by Kleckner, Scheeler and Irvine with a clover-shaped ring [K-S-I]. It is highly recommended to watch the video of their article, because the result they show in it is spectacular: the clover is first turned upside down, then turned upside down again, and so on. 12. The figure shows the mathematical simulation corresponding to the triangle, which is the same as that of the clover in the actual experiment. The video of this simulation https://sites.google.com/view/skumar1712/simulation-videos is available on the website [K].
fig. 12 itself shows the trajectory of a corner of the triangle drawn in blue. compared to the curve of Figure 11, it has a striking resemblance to the Riemann function! This shows that the function of Riemann has a physical and geometric structure of its own, as it appears in a natural experiment.
[Geometric study of Riemann's function]
Since the Riemann function represents a physical path, it is important to study its geometric and physical properties. This has been the focus of my doctoral thesis [E1]. Among the questions studied, I would highlight the following: if a particle follows this trajectory, is it possible to calculate the speed and direction of the particle at all points? The result is negative, due to the fact that the Riemann function is not derivable [E2]. But on the other hand, the particle moves with an average velocity and direction. Amazing, isn't it? To cite other results, the chances that the Hausdorff dimension of the trajectory is 4/3 are high [E3], and furthermore, from the turbulence theory side, the trajectory is intermittent [B-E-V].
There are a lot of mathematical questions that can be asked about Riemann’s function, non-derivable functions and fractals in general, all of which have great potential to describe natural events. Here is an excellent example of the work of many theoretical mathematicians, often mysterious and unknown!
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PICTURES AND IMAGES
Unless otherwise stated, the images are those created by the author of the article.
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THE BIBLIOGRAPHY
[B-E-V] A. Organized by Boritchev, D. From Eceizabarrena, V. Location of Vilaça da Rocha. Riemann's non-differentiable function is intermittent. Preprint (2019), arXiv:1910.13191, https://arxiv.org/abs/1910.13191
[E1] D. From the Eceizabarras. A geometric and physical study of Riemann’s non-differentiable function. PhD thesis (2020), University of the Basque Country (UPV/EHU). https://addi.ehu.es/handle/10810/49901
[E2] D. From the Eceizabarras. Geometric differentiability of Riemann’s non-differentiable function. I'm talking about Adv. I'm talking about Math. 366 (2020), 107091. https://doi.org/10.1016/j.aim.2020.107091
[E3] by D. From the Eceizabarras. On the Hausdorff dimension of Riemann’s non-differentiable function. Accepted in the journal Transactions of the American Mathematical Society. The Preprint: https://arxiv.org/abs/1910.02530
[G] J. It's about Gerver. The differentiability of the Riemann function at certain rational multiples of . I'm talking about Amer. According to J. I'm talking about Math. 92 (1970), 33-55. https://doi.org/10.2307/2373496
[H-V] F. de la Hoz, L. I'm talking about Vega. Vortex filament equation for a regular polygon. Nonlinearity 27 (2014), 3031–3057. https://doi.org/10.1088/0951-7715/27/12/3031
[K-S-I] D. More places to stay in Kleckner, M. I'm talking about W. According to Scheeler, W. I'm talking about T. M. I'm talking about Irvine. The life of a vortex knot. I'm talking about Phys. Fluids 26 (2014), 091105. https://doi.org/10.1063/1.4893590
[K] S. S. S. I'm talking about Kumar. The web page. https://sites.google.com/view/skumar1712/simulation-videos
[M] B. Assisted by Mandelbrot. The fractal geometry of nature. I'm talking about W. According to H. Images from Freeman and Co., In San Francisco, Calif., 1982.
[P] The H. I'm talking about Poincare. La logique et l’intuition dans la science mathématique et l’enseignement. In the Enseign. I'm talking about Math. 1 (1899), 157-162.
https://www.e-periodica.ch/digbib/view?pid=ens-001:1899:1#309
[W] Karl Weierstrass. Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzteren einen besetting in the Differentialente einen bestimmten. By Mathematische Werke. II. In Abhandlung 2. 1895, 71-74.
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