(a) That's correct. In Australia, statutory authorities are liable for positive acts in the same way as natural persons (although policy considerations may lower the duty of care owed by a statutory authority when competing considerations are taken into account). Statutory authorities can also be liable for omissions, particularly if a statute imposes a positive duty on the statutory authority to act. But in the present case, although the municipality has a statutory power to supervise, it is not obliged by the statute to do so.
However, a statutory authority may have a common law duty of care to a plaintiff as a member of an identifiable and vulnerable class of persons if harm is a reasonably foreseeable result of its acts or omissions, and if, on balancing all the relevant considerations, the court finds it is appropriate that a duty should be owed. In the circumstances of the case study, the municipality is likely to have owed a duty of care to B. Note that in some jurisdictions, legislation provides special provisions that define and limit the extent of the liability of a public authority. See, for example, Part 5 Civil Liability Act 2002 (NSW).
Note also that when a statutory authority is responsible for the control or management of land or a highway, it may be liable for omissions in the same way as other occupiers of land.