I read a result in a collected works of Steven Smale and one result leapt out at me which I'm clearly not understanding. Stated:
Theorem 1.1 (a): Suppose $J: M \to \mathbb{R}$ is a $C^2$ function, satisfies property (C) (see below), is bounded from below, and $M$ is a complete Riemannian manifold. Then the non-degenerate critical points of $J$ are isolated, and if on $J^{a,b}=J^{-1}[a,b]$ for $a,b$ finite, the critical points are non-degenerate, then the critical points of $J$ are finite in number.
And
Property (C) If $S$ is a subset of $M$ on which $|J|$ is bounded, but on which $\|J'(x)\|$ is not bounded away from zero, then there is a critical point of $J$ in the closure of $S$.
(Source.)
My question is how do I reconcile that with the function
$$J(x) = \int_0^x t^3 \cdot sin(1/t) dt$$
(where $J'(0)$ is defined to be zero.) As $x^3sin(1/x)$ on $[0,1]$ is $C^1$, bounded, and I believe satisfies (C), as I understand it the above theorem should say that the set of non-degenerate critical points of $J$ are finite on $J^{-1}[-1,1]$ which I don't believe they are.
Somewhere I'm clearly missing something, likely foolish, but any clarification would be greatly appreciated!
The point $0$ is a degenerate critical point, which you can verify by computing the second derivative.