I would like to study the convergence of the gradient descent method applied to the function
$$f(x)=|x-1|^3$$
In order to do that, I was thinking about using the following theorem:
Assume that $f : \mathbb{R}^n \to \mathbb{R}$ is convex and differentiable. Suppose the gradient of $f$ is Lipschitz continuous with constant $L>0$, meaning
$$\| \nabla f(x) - \nabla f(y) \| \leq L \| x-y \| $$
for any $x, y \in \mathbb{R}^n$. Then, the gradient descent with fixed step size $t \leq \frac{1}{L}$ satisfies
$$f(x^{(k)})-f(x^{*}) \leq \frac{\|x^{(0)}-x^{*}\|^2}{2 t k}$$
that is the gradient descent has convergence rate $O(\frac{1}{k})$.
My problem is that the gradient of the function I am studying is not Lipschitz, unless we put ourselves on a bounded interval, say $(1, a)$.
Could you help me? Any insight is very much appreciated :)