# Math Story & Creation II

Well, we already wrote in Part I about how the trigonometric series influenced the final statement of the concepts of function & continuity the way we know them today. But this is just the beginning!

In RIEMANN‘s Habilitation work (1854) he considered the inverse question related to the problem of the representation of a function by its Fourier Series, this is: If a function F is actually represented by a trigonometric series, how do their values F(x) behave by a continuous change in the argument x?…

As nothing is known about the function F in this case, and the coefficients of the Fourier series of the function are given by definite integrals, then he first had to precisely define what we understand by the definite integral of a function!!… So, he wrote an addendum where he introduced the now famous Riemann Integral and the integrability conditions.

This way –  thanks to these incredible series – concepts such as those of Function, Continuity or Integral were for the first time exactly and generally given to the math world!

And this is far not the end of the story … 😉 …

Riemann Integral – Regular Partition GIF

Riemann Integral – Irregular Partition GIF