On page 159, note 21, of Cauchy and the continuum, Imre Lakatos writes: "The modern definition of continuity [the $\epsilon-\delta$ definition] is strongly counter-intuitive, e.g. it is not invariant to rotation".
I have two questions:
- Does Lakatos mean that there is a continuous function $f$ such that ROTATION$(f)$ is not continuous?
- What are rotation and invariance to rotation in this context? (Lakatos does not explain the terms and a search on the internet did not clarify the issue.)
Thanks very much.
If the author did not further specify the meaning, then, in general, there is no way to determine the meaning intended. However, it is possible to make educated guesses. Given a function $\,f(x)\,$ on an interval, then the set of points $\,(x,f(x))\,$ forms a parametrized curve. This curve can be rotated. What the author probably meant is that there are some examples where the original curve is continuous, and a rotated version also comes from a function, but the rotated version is not continuous at some points. This all has to be made explicit, of course, but that is my guess.