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 
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)]](https://s0.wp.com/latex.php?latex=%28%5Cforall%5Cdelta%5Capprox+0%29%5B%7B+%7D%5E%2Af%28x%2B%5Cdelta%29%5Capprox+f%28x%29%2Bf%27%28x%29%5Cdelta%5D%5CRightarrow%28%5Cforall%5Cdelta%5Capprox+0%29%5B%7B+%7D%5E%2Af%28x%2B%5Cdelta%29%5Capprox+f%28x%29%5D&bg=ffffff&fg=333333&s=0 "(\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)]](https://s0.wp.com/latex.php?latex=%28%5Cforall%5Cdelta%5Capprox+0%29%5B%7B+%7D%5E%2Af%28x%2B%5Cdelta%29%5Capprox+f%28x%29%2Bf%27%28x%29%5Cdelta%5D%5Cvdash%28%5Cforall%5Cdelta%5Capprox+0%29%5B%7B+%7D%5E%2Af%28x%2B%5Cdelta%29%5Capprox+f%28x%29%5D&bg=ffffff&fg=333333&s=0 "(\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)]](https://s0.wp.com/latex.php?latex=%28%5Cforall%5Cdelta%5Capprox+0%29%5B%7B+%7D%5E%2Af%28x%2B%5Cdelta%29%5Capprox+f%28x%29%2Bf%27%28x%29%5Cdelta%5D%5CvDash%28%5Cforall%5Cdelta%5Capprox+0%29%5B%7B+%7D%5E%2Af%28x%2B%5Cdelta%29%5Capprox+f%28x%29%5D&bg=ffffff&fg=333333&s=0 "(\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 
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.
