Abstract
Let us try to find a function f with the following two properties.
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Notes
- 1.
It is by no means obvious that a power series can be differentiated term by term. A more precise analysis shows that we may rightfully do so, because \(\exp (x)\) converges for all \(x \in \mathbf{R}\) (radius of convergence = \(\infty \)) and therefore, the function obtained by term-by-term differentiation also converges for all \(x \in \mathbf{R}\).
- 2.
A piecewise continuous function is continuous except for possible isolated points of discontinuity. Continuity comprises piecewise continuity.
- 3.
Even in the literature one occasionally finds nonsensical \(\int _a^x f(x) dx\).
- 4.
From [32].
- 5.
Leonhard Euler (1707–1783) was one of the most prominent mathematicians of all times. He did groundbreaking work in calculus, number theory, and physics.
- 6.
An investigation is found in [6].
- 7.
Pierre Fermat (1601–1665) was a full-time lawyer. But he did seminal work in number theory, which was his passion. His famous conjecture that none of the integer equations \(a^n+b^n=c^n\) with \(n \ge 3\) has integer solutions has held many generations of mathematicians busy. Andrew Wiles succeeded in proving the conjecture of the century during the years around 1993. The book [28] tells this fascinating story brilliantly.
- 8.
With kind permission of Pixabay: This photograph is not subject to copyright. https://pixabay.com.
- 9.
This brainteaser is based on an oral communication by my friend Dr. ing. ETH Daniel Rufer, who in 2016 lost his life due to a bicycle accident. This problem is dedicated to him posthumously.
- 10.
I am obliged to dipl. math. P. Fässler for referring me to the clothoid.
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Fässler, A. (2019). Prerequisites from Calculus. In: Fast Track to Differential Equations. Springer, Cham. https://doi.org/10.1007/978-3-030-23291-7_1
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DOI: https://doi.org/10.1007/978-3-030-23291-7_1
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