Why is it necessary the first condition in Dirichlet Conditions (Fourier Series) In Wikipedia, it says that if a periodic function $f$ satisfies:
1- $f$ must be absolutely integrable over a period.
2- $f$ must be of bounded variation in any given bounded interval.
3- $f$ must have a finite number of discontinuities in any given bounded interval, and the discontinuities cannot be infinite,
then its Fourier series converges at every point where $f$ is continuous to the image under $f$ of those points, and converges to the midpoint for those point where $f$ is discontinuous.
Doesn't condition 3 implies 1?   I guess condition 3 means "jump discontinuity" by "discontinuities cannot be infinite", doesn't it?  If so, $f$ MUST satisfies condition 1.
What am I thinking wrong there?
Thanks
 A: To understand the logic behind Dirichlet's conditions, you need to know their purpose : For what reason these condition has been kept?
Well, the purpose of condition 1 is to check if the Fourier series coefficient does not blows up/down to infinity. Fourier series has been representing the functions presuming the fact that it's coefficients are "well-behaved" , i.e, finite; which is not the case everytime. A function satisfying condition 1 of Dirichlet is sure to have finite coefficients. Reference: Signal and System by Alan V Oppenheim and Alan Willsky
Similarly, the purpose of condition 2 is to check whether in a period, for finite number of x, the extremum is attained by the function f(x). You'll find better explanation of condition 2 in this answer : Dirichlet conditions for the convergence of Fourier series
The third condition: "f must have a finite number of discontinuities in any given bounded interval, and the discontinuities cannot be infinite" simply means there should be finite number of finite discontinuities.
For example, in square wave, we can see that there is finite number of discontinuities in a period (just 2 discontinuous points in a period) and the discontinuity also lies between 0 to 1 and is not infinite. Thus square wave satisfies the third condition.
Your sentence that condition 3 implies 1 is true in a sense that if the discontinuity blows up to infinity (condition 3) implies that the Fourier series coefficient of such a function is also infinite (condition 1). But that is not the only implication, there are many other reasons due to which coefficient of Fourier series might tend to infinity.
