25 March 2016

Integrals and primitives, the ultimate truth

On its most elementary level, an integral is defined for a continuous integrand. Yet nobody will hesitate to apply it to a function which is continuous except for one jump; just split up the interval and use additivity. Fine, but what about a whole sequence of discontinuities, or even more? Just how discontinuous can a function be before ceasing to have an integral? Here is the whole truth, integrability of a function defined in terms of its sole continuity.

Sure enough, the integral can then be given either as the sup of lower sums or the inf of upper sums, and from here on any calculus text takes over. BTW, this is as far as Riemann will take you. Anything more discontinuous has no Riemann integral. Define f(x)=1 if x is rational, and 0 elsewhere and you get a function which is discontinuous everywhere in [0,1]. It has no Riemann integral on [0,1], though it does have a Lebesgue integral.

So far, integration has nothing to do with differentiation, and even a giant like Archimedes did not surmise a relation between surface area and tangents, though he mastered both subjects like no one else before Newton. Yet in the seventeenth century the Fundamental Theorem of Calculus showed exactly this: to dress a list of all integrals, just make a list of all derivatives, and read it from right to left. The effectiveness of integration by primitives depends upon your definition of 'primitive'. On its most elementary level, F is called a primitive of f if F'=f. Simple, yes, but not very applicable. A function with a single jump, e.g., has no such 'simple primitives' because a derivative cannot have jumps. Here too, we'll give you the whole truth.

Geometrically, the Lipschitz property simply says that segments with endpoints on the graph have slopes bounded above and below, and that's all you have to check!

Now we're ready to expose integration by primitives in its full splendour.

Everything, proofs included (except some well known calculus results) takes two pages of elementary stuff (here or here). An essential tool is the Heine-Borel theorem, considered earlier (here) in a more general context.