## Category: Exercises and dialogues

7 December 2022
###### The Hilbert Hotel: a tale of two functions.
Can we exploit some apparent paradoxes of infinity to construct an injective function $$f(x)$$ defined over some closed and bounded interval, that is differentiable at some point...
5 January 2022
###### A bumpy ride
We take again inspiration from the Putnam competition, for an exercise on continuity and differentiability of real fuctions obtained by "stretching" a basic replica. Let...
5 December 2021
###### Problem unraveling
Let us go back to Terence Tao's Solving Mathematical Problems, for another exercise. As done in a previous post, I would like to show an alternative solution to a problem...
2 December 2021
###### Extreme derivatives
Theorem. Let $$f$$ be a real function, which is differentiable up to order $$n$$ in  $$c \in \Bbb R$$, for some positive integer $$n$$. Suppose that $$f^{(n)}(c) \neq 0$$,...
30 October 2021
###### Euclid strikes back
I found the following problem on Terenece Tao's "Solving Mathematical Problems". I propose, here, a synthetic approach based on simple Euclidean geometry concepts, instead of...
17 July 2021
###### Close enough?
While solving some exercises for my students, I bumped into the following sub-problem: given a divergent sequence $$(a_n)$$, such that $$(a_{n+1} - a_n)$$ converges to $$1$$, can...
25 September 2020
9 July 2020
###### Uneven integration
Very seldom does one need to make use of the actual definition of Riemann integral. Most of the times, in fact, we are dealing with very "regular" functions for which a whole...
4 July 2020
###### Continuing on continuity
We get back to real analysis and continuity in this post, to show the equivalence of two different definitions of continuity for real functions: the one based...
17 April 2020
###### "And I shall put discontinuity in thy domain"
I recently noticed on Youtube a solution to an indefinite integral that I think is a useful chance to clarify an important point about antidifferentiation of a given...
14 April 2020
###### Good things come in threes
We investigate in this post whether there is any statistical way of confirming or disproving this famous saying.  It is pretty obvious that, if we are tossing a fair...
3 March 2020
###### Coronavirus and logarithms
Here is a nice chance to practice on logarithms, addressed in particular to my high-school students, currently kept home, here in Milan, by our temporary health...
20 January 2020
###### E-piphany
Euler's number $$e$$, the base of the natural logarithm, manifests itself in various mathematical contexts, sometimes unexpectedly. In the following you will read an example of...
23 December 2019
###### Analysis and synthesis
While solving some geometrical problems, I have been investigating on the possible alternatives between a trigonometric approach versus an approach based on congruences and...
25 November 2019
###### Breaking the harmony
I found this curious exercise on the book Problems and Theorems in Analysis I, by Pólya and Szegö, and I thought it was a nice problem to propose here, since its solution...
14 November 2019
###### Non sequitur
In this post the behavior of sequences obtained by convolving two sequences is analyzed. In particular, we will give sufficient conditions for the convolution to be a null...
26 October 2019
###### Menelaus and infinitesimals
We show another application of Menalaus's Theorem, that, together with some infinitesimal calculus, will yield an unexpectedly simple result. Consider a square of side...
22 October 2019
###### The revenge of Menelaus
This post is devoted to Menelaus's Theorem, named for the astronomer Menelaus of Alexandria. It is a very imporant theorem in geometry, relating the ratios of the segments...
10 October 2019
###### My slant on asymptotes
In this post we will investigate the connection between slant asymptotes and the behavior of the derivative at $$\infty$$ for differentiable real functions. We say that a...