# Thread: Topology - Darboux Theorem

1. ## Topology - Darboux Theorem

Let I be an open interval in $\displaystyle \Re$ and let $\displaystyle f:I \rightarrow \Re$ be a differentiable function.

Let T be the set $\displaystyle T = \{(x,y) \in I \times I : x<y\}$. I can show that this is a connected subset of $\displaystyle \Re^2$ with the standard topology.

Let $\displaystyle g: T \rightarrow \Re$ be a function defined by $\displaystyle g(x,y) = (f(x) - f(y)) / (x-y)$. Prove that $\displaystyle g(T) \subset f'(I) \subset \overline{g(T)}$.

Show that $\displaystyle f'(I)$ is an interval.

This last deduction would be trivial if g(T) was a connected set, but I don't see why this should necessarily be true. (It would be true if g was a continuous function but we are not given this. Nevertheless, I think there may be a way of showing it).

2. Hi, if you want g(T) to be connect, isn't it sufficient to have g continuous on T?

3. Hi, if you want g(T) to be connected, isn't it sufficient to have g continuous on T?

4. Sure, but can you actually show this?!

5. The derivative of f may be discontinuous then!! hummm

6. I think this is the whole point of the Darboux Theorem. The derivative of f satisfies the intermediate value theorem even if f' is not continuous. In other words, there is nothing to stop f' being discontinuous.

7. So instead of the continuity, you may prove certain "convexity" of g !

8. Unless you can actually prove this and why it helps, then I'm not totally convinced by this idea!

9. I wanted to show something like the image by g of an interval in T is an interval in R.

But $\displaystyle h(x,y) = f(x) - f(y)$ is continuous on $\displaystyle I\times I$ and $\displaystyle \frac{1}{x-y}$ continuous on T. So why g shouldn't be continuous on T ?

10. Yes, to be honest this was my concern exactly, even though it seems to contradict the "point" of the question! (in that the derivative doesn't have to be continuous). I think it's because of the form of T...

11. But the derivative is obtained only at the limit where g could be not defined or discontinuous!