I am confuse on the deffinition for a critical point which is called isolated. Please tell me what the isolated critical point?
2026-03-26 06:03:44.1774505024
The isolated critical point!
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The set of critical points of a function $f : A \to B$ is $$ CP(f) = \{ x \in A \mid {\bf d}f_x \text{ is not surjective} \}, $$ where ${\bf d} f_x$ denotes the derivative of $f$ at the point $x$. A point $x$ is an isolated critical point of $f$ if there exists $\delta > 0$ such that $ CP(f) \cap \{ y\in A \text{ s.t. } |x-y| < \delta \} = \{x\}. $