Concept:
Oxidation state is usually an integer, but in some species (like mixed valence compounds), average oxidation state may be fractional.
Step 1: Check S$_4$O$_6^{2-}$ (tetrathionate ion):
Let oxidation state of S = x
\[
4x + 6(-2) = -2
\]
\[
4x - 12 = -2 \Rightarrow 4x = 10 \Rightarrow x = 2.5
\]
Step 2: Interpretation:
This fractional value indicates different sulfur atoms have different oxidation states.
Conclusion:
Hence, non-integer oxidation state exists.