Let $f: \mathbf{R^n} \to \mathbf{R}$ be differentiable, nonconvex function. The gradient descent iteration is given by $$x^{k+1} = x^{k} - t_k \nabla f(x^k)$$ To ensure $\lim_{k \to \infty} x^k$ exists, one condition for $f$ is that the critical point of $f$ is isolated, see paper http://proceedings.mlr.press/v49/lee16.pdf. The critical point is defined as $\left\{x\in \mathbf{R}^n : \nabla f(x) = 0\right\}$. A critical point $x$ is isolated if there is a neighborhood $U$ around $x$, and $x$ is the only critical point in $U$.
This condition is new to me. How to understand this condition? What type of function satisfies the isolate critical points condition?
