Answer:
Part a:
is continuous at the initial value (0,0) so due to Picardi theorem there exists an interval such that the IVP has a unique solution.
Part b:
is not continuous at the initial value (0,0) so due to Picardi theorem there does not exist an interval such that the IVP has a unique solution.
part c:
is continuous at the initial value (0,1) so due to Picardi theorem there exists an interval such that the IVP has a unique solution.
Step-by-step explanation:
Part a
as 
Let

Now derivative wrt y is given as

Finding continuity via the initial value
is continuous on
also
is also continuous on 
Also
is continuous at the initial value (0,0) so due to Picardi theorem there exists an interval such that the IVP has a unique solution.
Part b
as 
Let

Now derivative wrt y is given as

Finding continuity via the initial value
is continuous on
also
is also continuous on 
Also
is not continuous at the initial value (0,0) so due to Picardi theorem there does not exist an interval such that the IVP has a unique solution.
Part c
as 
Let

Now derivative wrt y is given as

Finding continuity via the initial value
is continuous on
also
is also continuous on
when 
Also
is continuous at the initial value (0,1) so due to Picardi theorem there exists an interval such that the IVP has a unique solution.