Let $f \in C[0,1]$ and $f(0)=f(1)$.
How do we prove $\exists a \in [0,1/2]$ such that $f(a)=f(a+1/2)$?
In fact, for every positive integer $n$, there is some $a$, such that $f(a) = f(a+\frac{1}{n})$.
For any other non-zero real $r$ (i.e not of the form $\frac{1}{n}$), there is a continuous fu...
Don't use "root" in Australia. It has a very different meaning.
The Virtuoso Pianist (Le Piano virtuose) by Charles-Louis Hanon, is a compilation of sixty exercises meant to train the pianist in speed, precision, agility, and strength of all of the fingers and flexibility in the wrists. First published in Boulogne, in 1873, The Virtuoso Pianist is Hanon's most well-known work, and is still widely used by piano instructors and pupils. However, the applicability of these nineteenth-century exercises has been questioned by some piano instructors today.
Overview
The exercises address common problems which could hamper the performance abilities of a stud...
