Equivalency of two statements of Dirichlet Fourier series conditions

Question

I am studying the Fourier series right now and saw two differently stated Dirichlet's Fourier series conditions.

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• Wolfram MathWorld says

A piecewise regular function that

1. Has a finite number of finite discontinuities
2. Has a finite number of extrema

can be expanded in a Fourier series which converges to the function at continuous points and the mean of the positive and negative limits at points of discontinuity.

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• WikiPedia says

The Dirichlet conditions are sufficient conditions for a real-valued, periodic function $f$ to be equal to the sum of its Fourier series at each point where $f$ is continuous. The conditions are

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.

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I can see that the number of discontinuities are restricted in the same way, but cannot really see how these two are equivalent conditions/definitions overall (if they really are).

Answer 1

You are correct that not both sets of conditions are "Dirichlet conditions." Wolfram is accurate. Wikipedia is wrong. The Dirichlet conditions would not have included anything about bounded variation because that concept came half a century after Dirichlet's formulation. Bounded variation was introduced by Jordan (of Jordan curve fame,) around 1880. Dirichlet died in 1851. Wikipedia entries are not scrutinized all that carefully.