I am looking for an example of a function $f: [0, 1] \longrightarrow \mathbb{R}$ that is in the Sobolev space of order $1/2$, $H^{1/2}([0, 1])$, but not in the Sobolev space of order $1/2 + \varepsilon$, for any $\varepsilon$.
Functions with discontinuities like the Rectangular function or the sawtooth wave are in the Sobolev spaces of order $s< 1/2$, but not equal to $s$.