Calculus: How To Really Understand A Function

A deep dive into one of the most fundamental math concepts!

Illustration created by the author (plots not to scale)

In calculus, arguably one of the most basic concepts is that of the function. The term ‘function’ was originally used for the first time by Gottfried Wilhelm Leibniz in 1673 to describe the “functional” properties of mathematical curves.

Over the years, this concept has evolved and expanded to become one of the most fundamental concepts in not just calculus but the whole of mathematics. In this essay, I will be diving into a deep discussion about how to really understand the notion of a function.

To this end, I will start by introducing its conventional use in continuous mathematics. Here, I will also be lingering upon some of the key technical characteristics and terms in relation to functions.

Then, I will proceed to expand the notion of a function to the world of discrete/discontinuous mathematics. Finally, I will try to generalise the concept as to how mathematics uses it today.

I’ve noticed that due to various technical and rhetorical reasons, young students are not exposed to the complete nature of functions. On the other hand, as students transform into advanced practitioners, they often get used to computational devices to handle ‘functions’ without having…