In the late 19th century, Karl Weierstrass invented a fractal-like function that was decried as nothing less than a “deplorable evil.” In time, it would transform the foundations of mathematics.

Derivatives: definitions, notation, and rules. A derivative is a function which measures the slope. It depends upon x in some way, and is found by differentiating a function of the form y = f (x).

At the same time, he could also prove that at every point, his new formal definition of the function’s derivative never had a finite value; it always “blew up” to infinity. In other words, continuity ...

The Squeeze Theorem is a very useful theorem which allows you to compute limits of some trickier functions. This applet allows you to explore the Squeeze Theorem visually. Here is another example.

Extending calculus to functions of multiple variables, this concept deals with partial derivatives, multiple integrals, and vector calculus. It’s crucial for understanding complex systems in physics, ...

Derivatives of functions are computed using differential calculus. A derivative in calculus is defined as “the rate of change of quantity y with respect to quantity x”. This sounds very similar to the ...

To complete the proof it suffices to prove that is not a mock theta function according to the modern definition. However, by theorem 1.2 of ref. 21, is a mock theta function with shadow proportional ...

This has led to an alternate definition of a mock theta function. ... GN Watson, The final problem: An account of the mock theta functions. J Lond Math Soc 2, 55–80 (1936). Crossref. Google Scholar. 2 ...