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Parenthetically, once they are established for polynomials, they are forced upon us for any reasonable understanding of differentiation because of the Weierstrass approximation theorem, for example. Also differentiation as an aspect of factoring becomes apparent.

Parenthetically, once they are established for polynomials, they are forced upon us for any reasonable understanding of differentiation, at least for uniformly differentiable because of the Weierstrass approximation theorem, for example. Also differentiation as an aspect of factoring becomes apparent.

By the way, an English translation of Karcher's lecture notes is in progress. I also heard today via facebook form Ursula Weiss who is a math professor in Germany (we were both graduate students at Brandeis in the late 1980s) that she has just finished translating the first chapter.

Added on 1/28/2011. I have just got an e-mail from Hermann Karcher. It says: "it was nice to hear from you again. I read Rokhlin's talk. Maybe one has to go even farther: Sometimes I think, everybody needs his own explanation, and a successful teacher is good in guessing what each individual child needs.I also read your paper. You won't be surprised that I am familiar with your arguments. I am now retired for 7 years. My last three semester beginners course was my most successful one. Today I attended the PhD colloquium of the student of a younger colleague. That student (and some of the younger people in the audience) had been in that last course of mine. They were happy to see me again and say how much fun those three semesters had been. For various reasons I was then in a situation where I could completely ignore, how the standard course in analysis proceeds. During the first semester I did my own stuff,ending up at continuous functions and their uniform convergence at the END of the first term. The next two semesters proceeded as usual - but all the fun we had came from building the foundations differently and the fun stayed with us. I wish you similar experiences. Don't try to convert too many grown ups, just enjoy teaching. Best regards --Hermann Karcher."