Differentiability Implies Continuity ~~ A Response to Part of Timothy Gowers post on the Cambridge Mathematical Tripos

Professor Timothy Gowers, of Cambridge University Department of Pure Mathematics and Mathematical Statistics, has posted some thoughts about the Cambridge Mathematical Tripos. He shows some model answers, and in his discussion of the “easy” part of the examination, he indicates that for students taking the tests, speed can be an over-riding concern. In one instance, Professor Gowers presents model answers to the problem of showing that for a real-valued function, differentiability implies continuity. I will propose a briefer, more intuitive, model answer for students who are interested in speeding up their progress through many parts of such a test. Here I shall give only four such examples from the Robinson/Zakon/Luxemburg, etc school of nonstandard analysis, each of which is a reasonable model answer to the same question, namely, “Show that for a function of one real variable, differentiability implies continuity.”

Model Answer 1:

Since f is differentiable, if \delta\approx0, then { }^*f(x+\delta)\approx f(x)+f'(x)\delta. Since the product of an infinitesimal with a real number is infinitesimal, it follows that if \delta\approx0, then { }^*f(x+\delta)\approx f(x), which is the nonstandard analysis criterion for continuity of the function f at the real number x.

Model Answer 2 (using primarily symbolic logic notation):

(\forall\delta\approx 0)[{ }^*f(x+\delta)\approx f(x)+f'(x)\delta]\Rightarrow(\forall\delta\approx 0)[{ }^*f(x+\delta)\approx f(x)]

Model Answer 3 (using primarily symbolic logic notation):

(\forall\delta\approx 0)[{ }^*f(x+\delta)\approx f(x)+f'(x)\delta]\vdash(\forall\delta\approx 0)[{ }^*f(x+\delta)\approx f(x)]

Model Answer 4 (using primarily symbolic logic notation):

(\forall\delta\approx 0)[{ }^*f(x+\delta)\approx f(x)+f'(x)\delta]\vDash(\forall\delta\approx 0)[{ }^*f(x+\delta)\approx f(x)]

Technically, Model Answer 3 is somewhat better than Model Answer 2, since it shows that the student is aware that one is proving a theorem from a set of assumptions, and the single turnstyle (the symbol \vdash) means, to a mathematical logician, that there is a proof from the assumptions on its left side, of the claim symbolized on its right side. However, Model Answer 4 is even a little better than Model Answer 3, because it demonstrates that the student is aware that the kinds of proofs Mathematicians do are semantic in nature, rather than syntactic. Specifically, the double turnstyle, \vDash, means to a mathematical logician that all models of analysis satisfying the assumptions on the left of the symbol must also satisfy the claim written on its right. All in all, Model Answer 1, while longer than any of the other three, may be considered a little better, merely because it also reminds the reader of the intuitively obvious (and easily provable) nonstandard analysis fact that the product of a number of “finite magnitude” with an infinitesimal, is always infinitesimal. But what examiner, upon seeing that a student understands the proper notation and usage of Mathematical Logic and its application to real analysis in the form of nonstandard models of analysis, by writing any of these four model answers or something reasonably close to one of them, is going to mark down such a student? Hopefully, no such examiner exists!

In all of these model answers, however, there is some work left to the reader, aka the examiner, because many of the axioms of (standard or nonstandard) analysis are not listed (in Model Answers 2, 3, or 4) to the left of the implication symbol or turnstyle, or included (in Model Answer 1) in the first sentence of the passage given as the answer. But note that in the model answers given by Professor Gower, many axioms of analysis are also not listed, as they are “presumed known” to both the examiner and the examinee. In a sense, this just means that the examiner is not requiring the test-taker to give a perfect, complete, completely rigorous, formal proof. There is then no reason not to accept these shorter nonstandard answers, which are no less rigorous and are no less complete than the standard ones that are typically accepted for full credit.

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