0
$\begingroup$

I'm looking for an example of a function g: $\mathbb R^3 \rightarrow \mathbb R$ that is discontinuous in every (x, $y_1, y_2) \in \mathbb R^3$, yet every initial value problem $$y' = g(x,y)$$ $$y(x_0)=y_0, (x_0, y_{01}, y_{02}) \in\mathbb R^3$$ has a unique solution in $\mathbb R$.

The first thing that came to mind was the Dirichlet function, but since it has neither derivative nor Riemann-integral it doesn't seem very helpful.

$\endgroup$
  • $\begingroup$ You say $g$ has $\mathbb R^3$ as a domain, but then you write $g(x,y)$. $\endgroup$ – Git Gud May 24 '14 at 9:19
  • $\begingroup$ That's what the assignment says. I suppose a second order differential equation is meant, interpreted as a two dimensional system. $\endgroup$ – red May 24 '14 at 9:31
1
$\begingroup$

Think of $\mathbb R^n$ as being filled (foliated) with integral curves of some (reasonable) ODE $\dot {\mathbf y} = \mathbf f(\mathbf y)$. On each integral curve, multiply $\mathbf f$ by a positive constant $c$. The integral curve will not change. The constants you use may be different for different integral curves, thus making $c\mathbf f$ discontinuous.

Concrete example: $\dot {\mathbf y} = \mathbf g(\mathbf y)$ where $\mathbf g(\mathbf y)=(1,0)$ if $y_2\in \mathbb Q$ and $\mathbf g(\mathbf y)=(2,0)$ if $y_2\notin \mathbb Q$.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.