Thread: Limit and Derivative proof

Let 0,1) \rightarrow \mathbb {R} " alt="f0,1) \rightarrow \mathbb {R} " /> be differentiable and let be Bounded. Show that exist.

Idea so far.

So if I define a sequence that converges to a point at 0, f would be able to able to map that into the co-domain, would that be the right start?

Thanks.

Originally Posted by tttcomrader

Let 0,1) \rightarrow \mathbb {R} " alt="f0,1) \rightarrow \mathbb {R} " /> be differentiable and let be founded. [I'll assume that should be "bounded".] Show that exist.

Idea so far.

So if I define a sequence that converges to a point at 0, f would be able to able to map that into the co-domain, would that be the right start?

Yes, that is a good way to start. Now prove that the sequence is Cauchy, and hence has a limit. Then you'll need to check that this limit is the same for all sequences .

(A quicker way to say the same thing is that a function with a bounded derivative is uniformly continuous; and if a function is uniformly continuous on a set A then it extends continuously to the closure of A.)

So in a differentiable function f, with f'(x) bounded, any sequence would be cauchy if it is mapped by f?

Originally Posted by tttcomrader

So in a differentiable function f, with f'(x) bounded, any sequence would be cauchy if it is mapped by f?

Yes, by the mean value theorem. If for all x, then for some ξ between x and y, and so . If is Cauchy then , from which it follows that is also Cauchy.

Okay, so I'm picking up from after I know that converges.

Now, suppose that .

Since f is differetiable on (0,1), it is also continuous on (0,1), implies that since .

So is this enough to show that exist and it is also equals to ?

Please look at Prof. Opalg’s second comment about uniform continuity.
A function with a bounded derivative is uniformly continuous.
Now go back to the last question I helped you with.
I think that was a lemma for this question.

let be given, and find , and suppose that the sequence converges to 0. Then such that whenever implies that

Since f is uniformly continuous, we have

So for all sequences that converges to 0, the sequences after being mapped by the function f would converge to f(0).

I hope this is right, suicide or not.

Thanks!!!