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Is there any way to optimize this iterative function?
Today I came across the following function:
$$
f(x_0,\,\dots,\,x_{N-1}) =
\begin{cases}
\infty, & \text{if $x_i=0$ for $i=0,\,\dots,\,N-1$}, \\
\min\left\{x_0,\,x_1,\,\dots,\,x_{N-1}\right\}, & \text{otherwise}.
\end{cases}
$$
Clearly, $f$ is not continuous and $f(x)=\infty$ for $x=0$. Clearly, $f$ is discontinuous for all $N \geq 2$.
However, intuitively it seems clear that $f$ is always bigger than $x_i$ for $i=0,\,\dots,\,N-1$, which means that $f$ is convergent iff $x_i=0$ for $i=0,\,\dots,\,N-1$.
Is this correct? If yes, how to show it?
A:
For $x_0 = 0$ your function is not a point function. Take for example $x_1=0,\,x_2=0,\,x_3 = 1$. Then your function is $0$ and the point is $1$.
A generalization is to consider the point function $g(x_0, x
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