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This paper extends previous work from arxiv:1702.05223, which shows that themain theorem of Morse theory holds for a large class of functions on singularspaces, where the function and the underlying singular space are required tosatisfy the five conditions explained in detail in the introduction toarxiv:1702.05223. The fourth of these conditions requires that the gradientflow of the function is well-behaved near the critical points, which is a verynatural condition, but difficult to explicitly check for examples without adetailed knowledge of the flow. In this paper we prove a general result showingthat the first three conditions always imply the fourth when the underlyingspace is locally compact. Moreover, if the function is proper and analytic thenthe first four conditions are all satisfied.

Local behaviour of the gradient flow of an analytic function near the unstable set of a critical point