Is it true that everything that isn't linear is concave or convex (or both)?

Question

In the conclusion of anti-fragile by Taleb, he claims that "everything non-linear is concave or convex, or both".

Is the statement general, and if not, what are its limitations? I suppose he is talking about continuous functions, although he doesn't mention it. If yes, can you provide a proof for the statement - or at least a convincing sketch.

EDIT (3): we established that the statement is false in general. It seems that by " both concave and convex", Taleb means that the function is alternatively concave or convex on disjoints sub-domains, like the sinus function - so it still wouldn't be linear (otherwise his statement is obviously void).

So let's assume we have a continuous nonlinear function over a bounded real-valued domain; is there a decomposition of the domain such that every sub-domain, arbitrarily small, is either concave or convex? Or is there a function which doesn't have this property?

Answer 1

The answer is basically given in the comments. I write them up just to give the question some closure.

It makes only sense to speak about convexity / concavity of a real function $f$ if the domain is an interval. So the question is, is there a system $\mathscr I$ of intervals such that $f$ restricted on each interval in $\mathscr I$ is either convex or concave.

If we allow $\mathscr I$ to be uncountable, we can select the intervals as singletons and the statement is trivially true.

If we require $\mathscr I$ to be countable, then there exists at least one $I\in \mathscr I$ with positive length. Now, a convex or concave function on an open interval is locally Lipschitz continuous and thus almost everywhere differentiable. So $f$ needs to be almost everywhere differentiable on the interior of $I$. However, there are nowhere differentiable yet continuous functions like the Weierstrass function. So the statement is in general wrong. (Aside the set of nowhere differentiable, continuous functions is dense).

If we additionally require that $f$ is differentiable, then $f'$ needs to be monotone and thus again almost everywhere differentiable on any open interval on which $f$ is convex or concave. So we should at least require twice differentiability. It reduces to following statement / claim (think $g=f''$):

Let $g:[a,b]\to \mathbb R$ have the intermediate value property. Then, there exists a countable system of intervals $\mathscr I$ such that $g$ restricted on each $I\in\mathscr I$ is either completely non-negative or completely non-positive.

However, I have no idea whether it is true or not....