# Dirichlet Conditions

What is meaning of a signal having a "finite number of maxima and minima during any single period of time"?

• Is the given solution clear enough? – Laurent Duval Jul 16 '17 at 15:00

$$f(t)=t^2 \cos (1/t)$$
with $f(0)=0$ by continuity. The ripples around $0$ get narrower and more numerous, and are infinitely many, in any non empty interval containing $0$.
The function is evidently continuous, even differentiable. On each interval defined by $$\left[\frac{1}{(k+1)\pi+\pi/2},\frac{1}{k\pi+\pi/2}\right]$$ the function $f$ vanishes at both ends, and since not identically $0$, it attains at least one extremum on this interval. Hence the function has an infinity of extrema on each interval $[0,a]$, $a>0$. The exact locations of the extremas can be computed with the derivative of $f$.