Hi everyone I was wondering if anybody could help me with this question; I have done the first 2 parts and a seperate 4th part but part 3 is causing me mathematical pain.
Suppose that
satisfies .
1) Find f(0).
[easy to show that f(0) = 0 ]
2) Show that
[can be done easily by induction]
Now suppose that f is differentiable at 0.
3) Show that f(t) = t f'(0) for all [Hint: For fixed t, consider the sequence
OK, we know that f is differentiable at 0, so
and for a fixed t, the above sequence simplifies to which is a constant sequence, but how does this help me? I guess I can say that
Any help would be appreciated
No, it is not formally correct. Your problem is that you are mixing t as variable in the definition of derivative and t as a fixed value. Let say, if f is differentiable at 0 then
This implies that for any zero sequence we have
Hence, for a fixed t, you can take as a particular zero sequence and apply the hypothesis of differentiability at 0 of to conclude
But as you stated in your post you are "discretisnig" the definition of derivative in a confusing way. For the convergence of the limit of a particular sequence you can't deduce differentiability, you need the existence of the limit for all the possible zero sequences.