Can there exist a unique solution to an initial value problem if the hypotheses of the existence and uniqueness theorem are not satisfied?

Question

I have been thinking about this question for a while. I haven't found a definite answer, but I am led to believe that there can be a unique solution to an IVP outside of interval of validity. I just fail to prove it.

Answer 1

For example, let $$ f(x,y) = \cases{1 & if $xy < 0$\cr -1 & if $xy > 0$\cr 0 & if $xy = 0$\cr}$$ and consider the initial value problem $$ \eqalign{ \dfrac{dy}{dx} &= f(x,y) \cr y(0) &= 0\cr} $$ The hypotheses of the Existence and Uniqueness Theorem are not satisfied because $f(x,y)$ is not continuous in a neighbourhood of $(0,0)$, but there is a unique solution, namely $y=0$.