Find a pointlike function that does not attain its minimum value. Deduce that it's not of the type $u:x\mapsto \delta_z(x)+k$ where $z\in X$ and $k\in \mathbb{R}^+$.
The problem is from the book "Metric Spaces" from the author Micheal Searcoid.
My effort: In the metric space $(X,e)$, where $e$ is the usual metric, we produce the function $u:X\rightarrow \mathbb{R}^+$, with $u(x)=|x|=e(x,0)=\delta_0(x)$.
$u$ does not get the minimum value which is $0$.
Is this correct ?
Let (S,d) be an unbounded metric space.
Assume a in S.
Define f(x) = 1/d(a,x) if x /= a, = 1 otherwise.
Does f have a minimum?
Does g(x) = -d(a,x) have a minimum?
Let S be infinite and A = { x(n) : n in N } subset S.
Define f:S -> R, f(x(n)) = 1/n, f(x) = 1 if x not in A.
Does this f have a minimum?
For a surprise, consider the case when S is finite.