I Is the Fermi-Dirac distribution equal to zero at the state of highest energy?

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The Fermi-Dirac distribution approaches zero as energy levels increase, indicating that states at high energy are unoccupied at finite temperatures. At absolute zero, the distribution is zero for energies above the Fermi energy, confirming that these states are not occupied. Therefore, while the distribution being zero suggests no occupation, it does not necessarily equate to being the highest energy state. The highest energy state is defined by the Fermi energy, beyond which the probability of occupation is zero. Understanding these principles is crucial in quantum statistics and solid-state physics.
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Fermi-Dirac distribution
I`m sorry if this seems too obvious, just trying to clarify something. When Fermi-Dirac distribution is equal to zero , can we assume it is the state of

the highest energy? (Because the propability of occupation is zero)
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##f(\epsilon)## goes to zero as ##\epsilon \rightarrow \infty## for a finite temperature (##T>0##).

At absolute zero, ##T=0##, then ##f(\epsilon > \epsilon_\mathrm{F}) = 0##, where ##\epsilon_\mathrm{F}## is the Fermi energy.
 
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