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Let $X\subset \mathbb R^n$ be a compact subset. I'm having troubles to find a counterexample of a bounded function $f:X\to\mathbb R$ which its oscillation doesn't reach a minimum value.

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    What is the question? – 2017-01-23
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    you mean that the function $f$ doesn't need to be continuous? – 2017-01-23
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    @copper.hat I'm looking for an example of a bounded function $X\to \mathbb R$ such that its oscillation doesn't reach a minimum value. Please see my edit. – 2017-01-23
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    @JorgeFernándezHidalgo Please see my edit – 2017-01-23
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    @JorgeFernándezHidalgo no, the function doesn't need to be continuous – 2017-01-23
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    @user409198 Then take any function $g$ that attains its minimum value exactly *once* at $x_{min} \in X$, and define $f=g$ on $X$ except at $f(x_{min})=g(x_{min})+1\,$. – 2017-01-23
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    @dxiv of course! thank you very much – 2017-01-23

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If your definition of oscillation at a point includes the value at the point, then you may define $f: [0,1] \rightarrow {\Bbb R}$ as follows: $$ f (x) = \left\{ \begin{array} {ll} 1 & {\rm if\;} x=0 \\ x & {\rm if\;} x\in (0,1]\cap {\Bbb Q} \\ 0 & {\rm if\;} x\in [0,1]\setminus {\Bbb Q} \end{array} \right. $$ $inf_{x\in [0,1]} \omega(f,x)=0$ is not attained.