Discrete calculus

Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. The word calculus is a Latin word, meaning originally "small pebble"; as such pebbles were used for calculation, the meaning of the word has evolved and today usually means a method of computation. Meanwhile, calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the study of continuous change.

Discrete calculus has two entry points, differential calculus and integral calculus. Differential calculus concerns incremental rates of change and the slopes of piece-wise linear curves. Integral calculus concerns accumulation of quantities and the areas under piece-wise constant curves. These two points of view are related to each other by the fundamental theorem of discrete calculus.

The study of the concepts of change starts with their discrete form. The development is dependent on a parameter, the increment ${\displaystyle \Delta x}$ of the independent variable. If we so choose, we can make the increment smaller and smaller and find the continuous counterparts of these concepts as limits. Informally, the limit of discrete calculus as ${\displaystyle \Delta x\to 0}$ is infinitesimal calculus. Even though it serves as a discrete underpinning of calculus, the main value of discrete calculus is in applications.