Karl Weierstrass’s “Pathological” function
In the late nineteenth century, Karl Weierstrass invented a function that was received as a “deplorable evil.” Over time, that “monstrous function” eventually transformed the foundations of mathematics.
Their fundamental concepts were rooted in intuition and informal arguments rather than precise formal definitions. Until, in response, two schools of thought emerged.
French mathematicians were more concerned with applying the calculation to physics problems. On the other hand, in the nineteenth century, German mathematicians began to lie down everything below. They began to look for counter-exposes that could have been supposedly defended assumptions and eventually used these counter-exemplos to put the calculation on a more stable and lasting base.
One of these mathematicians was Karl Weierstrass. Having given up on studying finances, Weierstrass began his career as a professional mathematician when he was almost 40 years old.
However, it was enough time to revolutionize this science, introducing one:
“Mathematical Monster”
In 1872, Weierstrass published a function that threatened everything that mathematicians thought to understand about the calculation.
Was received with indifference, anger and fearparticularly by the mathematical giants of the French School of Thought.
Henri Poincaréon the one hand, he named the new Weierstrass discovery as “An outrage to common sense”.
Charles Hermiteon the other hand, he called him a “Mal deplorable”.
To realize why Weierstrass’s result was so badly received, suggests that we begin by understanding two of the most fundamental concepts of calculation:
Continuity
A continuous function is exactly what it seems – a function that has no intervals or heels. It is possible to draw a path of any point in this function to any other without lifting the pencil.
Calculation is largely to determine how the continuous functions change. Works, in general terms, approaching a given function with straight and non -vertical lines.
Differentiability
At any point in this curve, it is possible to draw a “tangent” line – a line that best comes to the curve near that point. The slope, or slope, of the tangent line measures how quickly the function is changing at this point. It can define another function, called derivative, which provides the slope of the tangent line at each point of its original function. If the derivative exists at all points, the original function is said to be differentiated.
Functions that contain discontinuities are never differentiated.
That is, it will not be possible to draw a tangent line that approaches the discontinuities, which means that your derivative will not exist there. But even continuous functions are not always differential at all points.
This did not disturb most nineteenth -century mathematicians. They saw him as an isolated phenomenon. As long as the function is continuous, they said, there can only be a finite number of points where the derivative is not defined. At all other points, the function must remain pleasant and smooth. In other words, a function can only make zigzags and zagues to a certain point.
Or maybe not …
In fact, in 1806, a prominent French mathematician, André-Marie Ampèrehe said he had proved this fact.
For decades, its reasoning has not been concerned… until the “Monrist the weirstone”.
Weierstrass discovered a function that, according to the Ampère test, should be impossible: Continuous everywhere and not differentiated anywhere. Built it infinitely adding many wave “cosine” functions.
“Pathological function”? It cost to believe
Many mathematicians rejected the function. “It was an anomaly,” they said. Furthermore, could not even visualize it. In fact, at first glance, when trying to draw the graph of the Weierstrass function, it seems continuous in certain regions.
Only with zoom do you see that these regions are also “seriled” and that they will continue to get even more seriled and poorly behaved – what the mathematicians call “pathological” – With each additional expansion.
But Weierstrass had proved without margin for doubt That, although its function had no discontinuities, it was never differentiated.
To demonstrate it, it began by reviewing the definitions of “continuity” and “differentiability” that had been formulated decades earlier by mathematicians Augustine-Louis Cauchy and Bernard Bolzano.
As Quanta Magazine writes, these definitions were based on vague descriptions, in simple language and an inconsistent notation, which made them easy to interpret incorrectly. Therefore, Weierstrass rewrote them, using precise language and concrete mathematical formulas.
With this, it was able to prove that, at each point, its new formal definition of the derivative of the function never had a finite value; “He burst” always to the infinity.
By the right words: continuity did not imply the differentiability.
And his function was really monstrous as mathematicians had “fear.” As “punishment”, they were forced to follow the weierstrass footsteps.
Weierstrass revealed that Mathematics is full of monsters: Functions that seem impossible, foreign objects, wild behaviors, etc.; And its function has eventually had many practical applications.
In the early twentieth century, physicists wanted Study the Brownian Movementthe random movement of particles in a liquid or gas. Since this movement is continuous but not smooth – characterized by fast and infinitely small fluctuations – functions such as Weierstrass’s were perfect for modeling.
Similarly, these functions have been used to model uncertainty in the way people make decisions and assume risks, as well as the Behavior of Financial Markets.
