Consider the initial value problem
$$y'(t)=f\bigl(y(t)\bigr),\qquad y(0)=a \in \mathbf R$$
where $f\colon \mathbf R \to \mathbf R$.
Which of the following statements are necessarily true?
a) There exists a continuous function $f\colon \mathbf R \to \mathbf R$ and $a \in \mathbf R$ such that the above problem does not have a solution in any neighbourhood of $0$.
b) When $f$ is twice continuously differentiable, the maximal interval of existence for the above initial value problem is $\mathbf R$.