Examples of functions in $C(\mathbb{S}^1) \setminus H^{1/2}(\mathbb{S}^1) $

Question

Let $B$ be the open unit disk in $\mathbb{R}^2$. I was wondering about (possibly simple) examples of functions which are

1. in $C^2(B)\cap C^0(\bar B)$ but not in $H^{1/2}(\partial B)$;

and (possibly simple) examples of functions which are

2. in $C^0(\partial B)$ but not in $H^{1/2}(\partial B)$;

I know about the counterexample of Hadamard, but I was wondering about simpler examples (as I don't need the function to be harmonic in $B$).