Calculus: What It Is and How It Works

What calculus is

The Treccani Encyclopedia of Mathematics defines the field, in its entry on “analisi matematica,” as the branch that studies functions defined on sets of real or complex numbers. Around that definition sit a handful of central ideas — limit, derivative, integral, series, differential equations — that build on one another: first the function, then the limit, then continuity, then the derivative, and finally the integral.

The same entry splits the discipline into two main branches: real analysis, which studies real functions of a real variable, and complex analysis, which studies functions of a complex variable. Around these sit more specialized fields — harmonic analysis, nonstandard analysis, measure theory, functional analysis, differential equations — that fall outside the scope of this Recap; here the focus stays on the core, historically called infinitesimal analysis and known today, more often, simply as calculus.

For anyone coming from algebra and arithmetic, calculus isn’t a separate field so much as a further step: functions are written and manipulated with the same rules of symbolic notation, but the goal shifts, since the point is no longer solving an equation but describing how a quantity changes or accumulates.

Functions, the starting point

Wolfram MathWorld defines a function as a mapping f from a set A, the domain, to a set B, the codomain, such that every element of A is paired with exactly one element f(a) of B. The relationship can be many-to-one — several elements of the domain can share the same image — but never the reverse: a single element of the domain can’t point to two different outputs.

The word “function” in a mathematical sense first appears in a letter Leibniz wrote in August 1673, still with a fairly loose meaning. It was Johann Bernoulli, in a letter to Leibniz dated 2 September 1694, who gave it a more technical description, as a quantity built from variable and constant quantities. Euler sharpened it further: in his 1748 Introductio in analysin infinitorum he defined it as an analytic expression composed in any way from the variable and constants; by 1755, in the Institutiones calculi differentialis, he offered a broader version tied not to an explicit formula but to plain dependence between quantities — if one changes whenever another does, the first is a function of the second.

Limits, approaching without needing to arrive

A limit describes how a function behaves as the variable gets closer to a point, without requiring the function to be defined exactly there. In the formal “epsilon-delta” definition reported by MathWorld, a function has limit c at a point a if, for any margin of error chosen as small as one likes, there is a distance from the point within which the function’s values stay inside that margin. The same idea applies to sequences of numbers: a sequence has limit L if, from some index onward, every term stays as close to L as one wants.

It’s a concept that underpins everything else: continuity, the derivative, and the integral are all defined starting from a limit.

Continuity, when the graph doesn’t break

A function is continuous at a point x₀ when three conditions hold together: the function is defined at x₀, its limit at x₀ exists, and that limit matches the function’s value at x₀. If even one of these three conditions fails, the function has a discontinuity there — a jump, a gap, or an oscillation with no limit.

Continuity is a necessary condition for differentiability, but it isn’t sufficient: a function can be continuous at a point and still have no derivative there, if the difference quotient doesn’t approach the same value from the right and from the left.

The derivative, the speed of change

The derivative of a function at a point is the limit of the difference quotient [f(x+h) - f(x)] / h as h approaches zero: it measures how fast the function changes right at that point, not its average change over an interval.

Practical example: if a function describes an object’s position over time, its derivative describes that object’s instantaneous velocity at every moment — and it’s precisely in terms of instantaneous velocity and acceleration, that is, derivatives, that Newton’s three laws of motion are written.

Historically, the derivative had two parallel notations. Leibniz wrote d/dx; Newton used a dot above the letter to mark “fluxions,” instantaneous speeds — a notion he introduced as early as October 1666, in his Tract on Fluxions, alongside what MacTutor describes as the earliest explicit statement of the fundamental theorem of calculus. According to MathWorld, Leibniz’s notation eventually won out over Newton’s in everyday use.

The integral, adding up infinitely many pieces

The definite integral ∫ₐᵇ f(x)dx adds up infinitely many infinitesimal pieces to compute the content of a continuous region, typically an area under a function’s graph, between two limits a and b. The indefinite integral ∫f(x)dx, on the other hand, isn’t a number but a family of functions: it’s defined only up to an arbitrary constant C, because if F(x) is an antiderivative of f(x), then F(x)+C is one too, since the derivative of a constant is always zero.

The link between the two ideas is the fundamental theorem of calculus, whose first part — as reported by MathWorld — states that ∫ₐᵇ f(x)dx = F(b) - F(a), where F is an antiderivative of f: the derivative and the integral are, in a sense, inverse operations of one another.

A story two thousand years long

The core idea of adding up infinitely many tiny pieces predates the 1600s by a wide margin: according to MacTutor, Archimedes, around 225 BC, already computed the area of a parabolic segment using what the source describes as the earliest known example of summing an infinite series.

Calculus itself was born in the second half of the 1600s, when Newton and Leibniz arrived at similar results while working independently. Leibniz met Christiaan Huygens in Paris in 1672 and, the following year, bought mathematical works in London, including some by Isaac Barrow; in 1675 he wrote the first expression using the integral symbol, and he published his results on differential and integral calculus in the Acta Eruditorum in 1684 and 1686. Newton, for his part, had written Analysis with infinite series in 1669 and Method of fluxions in 1671, but only published them many years later, in 1711 and 1736 respectively; according to MacTutor, the failure of the publisher of one of Barrow’s works made publishers more cautious about printing mathematical texts for a period, a factor that contributed to the delay. It was Jacob Bernoulli, in 1690, who proposed the name “integral calculus” for the branch of the field previously called calculus summatorius.

The rigor with which limit and continuity are defined arrived later. In 1734 the philosopher George Berkeley published The Analyst, a work that attacked the lack of solid foundations in the calculus of the time; his criticism opened, over the following decades, a path that culminated in Cauchy’s nineteenth-century work on the rigorous definitions of limit and continuity that later became standard.