I can't find the counterexample of function with this criteria:
Function $f$ such that $f'$ is absolute continuous in $[a,b]$, $f'' \notin L^2[a,b]$ but $f''$ is bounded in $[a,b]$.
Function $f$ such that $f'$ is absolute continuous in $[a,b]$, $f'' \in L^2[a,b]$ but $f''$ is not bounded in $[a,b]$.