What does it mean for a function to be continuous?
Received wisdom is that this was sorted out in the nineteenth century by Cauchy and Weierstrass; even if the epsilon-delta definition is a bit of a mouthful for students, once they have mastered it there is no further problem.
But an interesting article by J. F. Harper in the most recent BSHM Bulletin (doi 10.1080/17498430.2015.1116053) looks at definitions used by Bolzano, Cauchy, Duhamel, Weierstrass, Heine, Jordan, Whittaker, Goursat, Pierpont, Hobson, Hardy, Hausdorff, Courant, Siddons, Rudin, Bartle and Sherbert, Carlson, Gowers, and Kudryatsev, and finds numerous inconsistencies, some of them between different works by the same author.
The procedure is to compare these authors’ definitions on four particular real functions:
- the function which is 0 at the origin and undefined elsewhere;
- the function which is 0 on the positive reals and undefined elsewhere;
- the function which is 0 on the rationals and undefined elsewhere;
- the function which is 0 at the origin and sin(1/x) at the non-zero real number x.
But hang on; the first three are not functions on the real numbers!
So there is a more basic confusion, about what is meant by a function.
I recommend to you Littlewood’s article “From Fermat’s Last Theorem to the abolition of capital punishment”, which appears in his Miscellany (a book you should certainly read!). Littlewood quotes, as an “intellectual treat”, the muddled ramblings of Forsyth, in his Theory of Functions of a Complex Variable, stretching over two pages. Littlewood goes on to say,
Nowadays, of course, a function y = y(x) means there is a class of “arguments” x, and to each x there is one and only one “value” y. After some trivial explanations (or none?) we can be balder still, and say that a function is a class C of pairs (x,y) (order within the bracket counting), C being subject (only) to the condition that the x‘s of different points are different.
Actually, even these two definitions are not equivalent; the second allows all four functions above, while the first forbids the first three.
We normally define a function f : X → Y. The correct logical definition is as a set of ordered pairs, but we should replace Littlewood’s second condition by the condition that each element of X occurs once and only once as the first component of an ordered pair in the set. In other words, the domain is part of the definition of a function.
I tend to explain this to students by saying that a function is a black box: you put in an element x, and out comes an element y. (Unlike earlier mathematicians, we neither know nor care how the black box actually operates; it is defined solely by the inputs and corresponding outputs). But, like all good black boxes, it comes which a guarantee, which states that if an element of X is put in, then out will come an element of Y. If you put in an element which does not belong to X, the behaviour is not guaranteed; anything might happen (but to agree with the logical definition, it is perhaps better to say that nothing comes out).
This definition has the disadvantage that the codomain is also part of the definition of the function. So the squaring function from the natural numbers to the real numbers is a different function from the squaring function from the natural numbers to the natural numbers.
I am not too troubled by this. After all, in the standard constructions of the number systems, the natural number 1, the integer 1, the rational number 1, the real number 1, and the complex number 1 are completely different objects; yet we use them as if they are all the same thing, without getting into trouble.
If the domain is part of the definition of a function, then Harper’s first three functions are not functions from the real numbers to the real numbers. I would be much more interested to have genuine examples of real functions which discriminated among proposed definitions of continuity.