Peano's Theorem Examples Peano's theorem for ordinary differential equations says that when the function F(x,u) is continuous on a square $S=[x_0-r,x_0+r]\times [ y_0-r,y_0+r]$ in 
$u' = F(x,u)$
then the IVP $u(x_0)=y_0$ has a solution on some region $[x_0-h,x_0+h]$ where 
$0<h\le r$
$h \max_S |F| \le r$
I understand the proof, but I have a real lack of good examples of ordinary differential equations.  Can anyone provide examples of the following that illustrate the assumptions of the theorem?
(0) A bounded discontinuous function $F$ that has no solution to some initial value problem
(1) An example of a function $F$ that's continuous on a square $S$ such that $F$ has a solution on a subset of the square but does not have a solution on the entire square?
(2) A function $F$ that is continuous everywhere that doesn't have a global solution to some initial value problem.  I can't tell from the theorem whether such an example would exist or not, because potentially your largest choice of $h$ might converge to a finite value as  $r$ gets larger.  
Any other good examples that illuminate the theorem are welcome.  
 A: Here are some examples.


*

*A bounded discontinuous function $F$ that has no solution to some
initial value problem


Take $$\begin{array}{l|rcl}
F : & \mathbb [-1,+1] \times [-1,+1] & \longrightarrow & \mathbb R \\
    & (x,u) & \longmapsto & 1 \text{ if } x \in \mathbb Q \\
    & (x,u) & \longmapsto & 0 \text{ otherwise} \end{array}$$
According to Darboux Theorem a solution $u$ should have a derivative $u^\prime$ taking all values in segment $[0,1]$ which is in contradiction with the IVP.


*

*An example of a function $F$ that's continuous on a square $S$ such
that $F$ has a solution on a subset of the square but does not have a
solution on the entire square?


$$\begin{array}{l|rcl}
F : & \mathbb [-1,+1] \times [-1,+1] & \longrightarrow & \mathbb R \\
    & (x,u) & \longmapsto & u^2 \end{array}$$
Has for solution $u(x)=\frac{2}{1-2x}$ to the IVP $u(0)=2$ and the solution is not defined for $x \ge \frac{1}{2}$.


*

*A function $F$ that is continuous everywhere that doesn't have a
global solution to some initial value problem.


Have a look here. You need to look at infinite dimensional spaces to find a counterexample.
