Consider the I.V.P. $$\frac{dy}{dx}=f(y(x)),y(0)=a\in\mathbb{R}$$ where $f:\mathbb{R}\rightarrow\mathbb{R}.$ which of the following statements are necessarily true?
$1.$ There exist a continuous function $f$ and a real number $a$ such that the above problem does not have a solution in any neighbourhood of zero.
$2$. The above problem has a unique solution for ever real $a$ when $f$ is Lipschitz continuous.
$3.$ When$f$ is twice continuously differentiale the maximum interval of existence for the above problem is $\mathbb{R}$.
$4.$ The maximum interval of existence for the above problem is $\mathbb{R}$ when $f$ is bounded and continuously differentiable.
According to me by Picard existence and uniqueness theorem first option is false and second one is true. I have no result regarding existence of solution in whole real line $\mathbb{R}$. For option number three I have counter example if $y =y^{2}$ .Please suggest me some result regarding this so that I can understand last option. Thanks a lot.