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.

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Dowland's not-love song

Shortly before and shortly after the year 1600, English renaissance composer John Dowland published three Books of Songs. The third and last of these contains, as song number VII, the well known Say, Love. You can inspect it here, along with the first three pages of the book. It is popular among collections of love songs, as it should with such a title. Most of the time, a judicious choice is made from the four stanzas. Nevertheless, there is no concealing the problem: it is hard to make sense of it as a love poem. Lovers are not known for their logical consistency, but hey, there are limits.

A virgin, moon goddess, fairy queen, archer and rare mirror in one. The answer to the problem is simple once you have seen it: this is not a love song at all. It is a flattering ode to Elizabeth Tudor, the virgin queen, after whom Virginia was named. Though she had lovers, she never married, and could rejoice in the idolatrous admiration of the protestant part of her subjects. Her coat of arms was

with Latin motto Always the same. There were several concurring reasons why she was allegorically compared to the moon goddess Diana "Cynthia". Both were virgins (virgin queen, virgin goddess) and the moon reflected well the queen's motto: apparently changing, yet ever the same heavenly body. (As for 'reflected': yes, the moon is a mirror.) Painters, poets and playwrights alike used the moon goddess image. In literature we can mention Walter Rale(i)gh with The Ocean's Love to Cynthia, Ben Johnson with Cynthia's revels and sonnets by the unfortunate Robert Devereux, earl of Essex. And here is an early portrait of Elizabeth as Diana. Apart from the crescent, bow and arrows are also emblems of Diana, who was goddess of the hunt as well.

Elizabeth as Diana, ca. 1560

Besides being identified with a goddess, Elizabeth was also unmistakably the main character in Spencer's long poem The Faerie Queene (1590 and 1596). We quote some verses from Book I, Canto VI, because the rare mirror also turns up in Dowland's song. (Here and below we have modernized the spelling.) 

They drawing nigh, unto their God present
That flower of faith and beauty excellent:
The God himself viewing that mirror rare,
Stood long amazed, and burnt in his intent;

That mirror rare, flower of faith and beauty excellent is, of course, Elizabeth in allegory.

Dowland's song. Dowland (or his lyricist if there is one) not only combined all these elements, but added another, which is the theme and final image of the poem: that Elizabeth's hunter's arrow and bow are in fact the arms of Amor, which he has handed over after unsuccessfully attacking a determined virgin-moon goddess-fairy queen guided (most unwomanly) by a constant mind. The poem is a dialogue between the poet and Amor (Love). To improve readability, we have put Amor's answers in italics.

Say, Love, if ever thou didst find
a woman with a constant mind?
    None but one.
And what should that rare mirror be?
   Some goddess or some queen is she,
   she and only she,
   she only queen of love and beauty.

But could thy fiery poisoned dart
at no time touch her spotless heart
nor come near?
   She is not subject to Love's bow.
   Her eye commands, her heart says 'no',
   'no' and only 'no',
   one 'no' another still does follow.

How might I that fair wonder know
That mocks desire with endless 'no'?
   See the moon,
   that ever in one change does grow
   yet still the same, and she is so,
   so and only so.
   From Heaven her virtues she does borrow.

To her then yield thy shafts and bow,
that can command affections so.
Love is free,
so are her thoughts that vanquish thee.
There is no queen of love but she,
she and only she,
she only queen of love and beauty. 

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