Question: From Numerical Optimization by Nocedal and Wright: 

> 
> A point $x^\*$ is a strict local minimizer if there is a neighborhood
>  $N$ of $x^\*$ such that $f(x^\*)<f(x)$ for all $x \in N$ with $x \neq
> > x^\*$.
> 
> 
> A point $x^\*$ is an isolated local minimizer if there is a
>  neighborhood $N$ of $x^\*$ such that $x^\*$ş is the only local minimizer
>  in $N$.
> 
> 
> ... 
> 
> 
> While strict local minimizers are not always isolated, it is true that
>  all isolated local minimizers are strict.
> 
> 
> 

I just could not draw an image of a strict local minimizer that is not isolated in my mind. Neither could I find in the web.

Can you draw one for me, so that I can see the different between strict and isolated local minimizers visually?

By the way authors give following function as an example: $f(x)=x^4cos(1/x)+2x^4, 0 $ being the minimizer. Though you do not need to show on this function. A hand drawn image is OK, if it shows the difference.

Answer: 
To get some function that looks like the graph in Maziar Sanjabi's answer, we can

1. Find a function that has a basic shape like that, such as $f(x)=x^2$;
2. Multiply $f(x)$ with a periodic function that has infinitely small period as it approaches $x=0$, which is the local minimizer of $f(x)$, one such function is $g(x)=2-\cos(30/x)$.

We have $h(x)=f(x)g(x)$, which looks like: 

[![enter image description here](https://i.stack.imgur.com/IM2QT.png)](https://i.stack.imgur.com/IM2QT.png) 

It oscillates between $x^2$ and $3x^2$.
Bertsekas's book "Nonlinear Programming" 1999 version (version 2), practice problem 1.1.11, gives another example that isolated local mininum is not strict local minimum (due to a paper [Dun87]).

Importantly, it also proves that non-singular isolated local minimum must be strict local minimum. Thus the example only happens to the singular case. This might help a bit on understanding when this would happen.