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Suppose I have a continuous function $h : \mathbb{R}^n \rightarrow \mathbb{R}$, and suppose $\mathbf{z}$ is a local minimum.

Why does it then imply that $\nabla h (\mathbf{z}) = \mathbf{0}$?

ps I apologize for the confusion. I forgot to mention that $\nabla h$ is well defined on $\mathbb{R}^n$.

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closed as off-topic by Guy Fsone, Namaste, Hans Lundmark, jgon, kimchi lover Jan 11 '18 at 23:44

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  • $\begingroup$ Relative Extrema are related to critical points. The critical point is where the gradient equals $0$. $\endgroup$ – Biggs Jan 11 '18 at 18:25
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    $\begingroup$ Note that $z=\sqrt{x^2+y^2}$ has a local minimum at $(0,0)$. But what is the gradient here? $\endgroup$ – Emilio Novati Jan 11 '18 at 18:27
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Just fix $e_i (0,0....1,0...0)$ a unit vector and define $g: (-1,1) ~~t\mapsto g(t) = h(z+te)$ then $t=0$ is a local minimum of the function $g$ is some small neighborhood of $t=0$.

Hence, $$0=g'(0)= \nabla h(z)\cdot e_i ~~~$$

That is for all $e_i$ we have $$ \nabla h(z)\cdot e_i =\frac{\partial h}{\partial x_i}(z)=0 $$

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    $\begingroup$ All we know is that $h$ is continuous, so this doesn't work. $\endgroup$ – zhw. Jan 11 '18 at 19:01
  • $\begingroup$ @zhw. can we talk about partial derivate of continuous functions?. The differentiability is implicitly assumed here. Please read the title $\endgroup$ – Guy Fsone Jan 11 '18 at 19:23
  • $\begingroup$ @zhw. what is the gradient of continuous functions? thanks very much for your down vote $\endgroup$ – Guy Fsone Jan 11 '18 at 19:28
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    $\begingroup$ No, the title does not imply that differentiability is assumed, as noted by others here. Even if we assume the partial derivatives exist, your proof for a an arbitrary unit vector $e$ is incorrect, as those directional derivatives may not exist. $\endgroup$ – zhw. Jan 11 '18 at 19:59
  • $\begingroup$ @zhw. hahah funny why dont you take $e=e_i$ this what what I meant $\endgroup$ – Guy Fsone Jan 11 '18 at 20:09
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Due to Emilio Novati's counterexample, let's first assume that the partial derivatives of $h$ exist in $\mathbf{z}$.

If we had that $\nabla h (\mathbf{z}) \neq \mathbf{0}$, then at least one of the partial derivatives would be non-zero, which means that function $h$ is increasing/decreasing in $\mathbf{z}$ with respect to that variable (if it's increasing, just let the variable vary the other way around, so it becomes decreasing), and thus as it is decreasing in $\mathbf{z}$, you cannot find a neighborhood in $\mathbf{z}$ where all $f(\mathbf{a})\geq f(\mathbf{z})$.

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