Calculus without limits # Critical points

Assume f is a continuous function of one real variable. Lets call a point p a critical point of f if Df(p)=0 where Df(x) = f(x+1)-f(x) is the discrete derivative of f. As in classical calculus, a point p is called a local maximum of f, if there exists an open neighborhood U of a, such that $f(p) \geq f(x)$ for all x in U. We define local minimum similarly. We can now state the following theorem:

Theorem: If a continuous function has a critical point p, then it has a local maximum or local minimum in the open interval $(p,p+1)$.

The proof is a simple consequence of the extreme value theorem. If f is smooth, then by the intermediate value theorem there is even a point in the open interval where we have a classical critical point.

Despite its offending simplicity, the theorem shows that with the right definitions, a theorem can become elegant. This theorem is of course false in classical calculus: the function $f(x) = x^3$ which has a critical point at 0 but no local maximum there.

The above theorem is not only unconditionally true, it is also more robust than the classical result. The only non-robust case is if f has a critical point at p and is constant between p and p+1. In the classical setup, any horizontal inflection point is structurally unstable. The smallest perturbation of the function can either make it monotone like with $f(x) = x^3 + ax$ where a is arbitrarily small and positive or can produce two local extrema when a is arbitrarily small but negative.

What about practical uses of this theorem? Lets go back to our DJI data which we have already seen in the Taylor page.