A question of the complete metric space!

1. Oct 7, 2008

simpleeyelid

Continuously differentiable Function $$C^1 {}$$ $$\left[0,1\right]$$ is complete with respect to the metric space
$$D_\infty{}$${f,g}=sup{$$\left|f(t)-g(t)\right|$$}+sup{$$\left|f^1{}(t)-g^1{}(t)\right|$$}

but not in the $$d_\infty{}$${f,g}=sup{$$\left|f(t)-g(t)\right|$$}

Thanks for the helps in advance.

Regards...

BI

Last edited: Oct 7, 2008
2. Oct 7, 2008

cogito²

For the counterexample, can you come up with a sequence of differentiable functions that converges to a non-differentiable function? Hint: Choose a very simple function that's continuous, but not differentiable.

3. Oct 7, 2008

simpleeyelid

yes, that is what I am trying to do, however, it is strange that, some theorem in the book "Analysis and Mathematical physics" says the 2nd holds in C[a,b] wrt the second distance function, I am now quite confused...

This is an original question from the book real analysis with economic applications.

Last edited: Oct 7, 2008
4. Oct 7, 2008

cogito²

The space of continuous functions is complete with respect to the second distance function. But what you're trying to show is that the space of continuously differentiable functions is not complete with respect to that norm. Since every continuously differentiable function is continuous, you know that the limit of any sequence of continuously differentiable functions has to be continuous. The problem is asking you to find sequence of differentiable functions whose limit (which is continuous) is not differentiable.

5. Oct 7, 2008