Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $1>a > 0$ and $x(0) = 0.$ Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $1>a > 0$ and $x(0) = 0.$
First, I notice that this equation is not differentiable at x = 0. Therefore, the solution is not unique. 
For existence, I am told to check whether there they are continuous. Could anyone specify the different cases that they are continuous?
Let $a = \frac{1}{x}$.
If x is even, then they are not continuous ...so we cannot guarantee there is a solution.
If x is odd, then they are continuous...so solutions exist.
Could anyone also check if these statements are correct?

The initial value problem
$$
x'=|x|^a, \quad x(0)=0,
$$
where $a\in (0,1)$ has infinitely many solutions, and it is the most typical example of violation of uniqueness in ODEs. Two of them are
$$
\psi_1(t)\equiv 0 \quad \&\quad \psi_2(t)=\left(\frac{t}{1-a}\right)^\frac{1}{1-a}\mathrm{sgn}\,t.
$$
Note. On the other hand the IVP
$$
x'=|x|^a+1, \quad x(0)=0,
$$
enjoys global existence and uniqueness, although $|x|^a+1$ is not Lipschitz continuous. Hence Lipschitz continuity is NOT NECESSARY for uniqueness.
