tensorgami

This note is deliberately not about giving the shortest proof of the theorem.

For this particular result, there is a very quick route. One can use the identity

max(f,g) = (f + g + |f − g|)/2

and then appeal to the standard closure properties of Riemann integrable functions. That proof is clean, efficient, and in many contexts exactly the right proof to use.

But that is not the purpose of this note.

The purpose is to open the hood. The identity above is a compression of a mechanism. It tells us that the result is true, but it does not force us to see why the local oscillation of max(f,g) is controlled by the local oscillations of f and g. In other words, it proves the theorem while hiding the gears.

This distinction matters. If one is merely using the theorem as a tool, the compressed proof is enough. But if one is trying to develop real analysis instincts, especially the kind needed to produce arguments rather than merely consume them, then it is a mistake to always retreat to the most elegant formulation as soon as it becomes available.

Finished mathematics often presents itself in polished form: a slick identity, a well-chosen abstraction, a clean theorem that absorbs all the messy cases. But at the frontier, one usually does not begin with such a formulation. One begins with complicated expressions, interacting terms, bad-looking cross terms, exceptional regimes, and estimates that have to be built by hand. The elegant structure, if it exists, is often discovered only after one has already done the rough work.

That is why this elementary theorem is a useful training ground. The function max(f,g) is simple enough that the entire mechanism can be seen, but nontrivial enough that there is a genuine mixed case to handle. On a small interval, the near-largest value of max(f,g) may come from f, while the near-smallest value may come from g. This creates a term of the form

f(αᵢ) − g(βᵢ).

At first glance, this is not the oscillation of f and it is not the oscillation of g. It is a mixed term. The proof has to use the order structure of the maximum to convert that mixed term into a same-function oscillation estimate. Namely, because g(βᵢ) is the maximum at βᵢ, we know that

g(βᵢ) ≥ f(βᵢ),

and therefore

f(αᵢ) − g(βᵢ) ≤ f(αᵢ) − f(βᵢ).

That small inequality is the point of the proof. It is a toy version of a much more general research maneuver: take a bad-looking interaction term and find the hidden structure that lets you charge it to something already under control.

So the note below should not be read as an attempt to outdo the one-line proof. It should be read as a proof workout. The theorem is elementary, but the proof pattern is serious: localize to intervals, choose near-extremal witnesses, split into cases, control the mixed terms, and then sum the local estimates globally.

There are two kinds of mathematical maturity involved here. One is compression maturity: knowing the right identity, abstraction, or theorem that makes an argument short. The other is production maturity: being able to operate before the right abstraction is visible. This note is mainly about the second kind.

Elegant proofs are valuable. But elegance is most powerful when it is earned, not when it is used as a way to avoid contact with the mechanism. The direct Darboux proof below is intentionally hands-on: it exposes the gears that the elegant proof hides.