# Thread: how to show that the following equality holds.

1. ## how to show that the following equality holds.

Show that $f(x)$ is the derivative of $f(.)$ at $x$ if and only if $lim_{h \to 0} \sup_{|t|\leqslant h} \frac{|f(x+t)-f(x)-tf^{'} (x)|}{h} = 0$

2. ## Re: how to show that the following equality holds.

Hey ujgilani.

Have you considered the triangle inequality?

3. ## Re: how to show that the following equality holds.

Hi, thanks for your reply
no I didn't try doing it with triangle inequality. could you please brief a little bit more.

4. ## Re: how to show that the following equality holds.

Basically the triangle inequality says |a+b| <= |a| + |b| where |.| is some norm.

5. ## Re: how to show that the following equality holds.

yeah i know the "triangle inequality" concept but question is how we can apply it here

6. ## Re: how to show that the following equality holds.

Show that |a| + |b| = 0.

7. ## Re: how to show that the following equality holds.

sorry no idea how to use it in this setup

8. ## Re: how to show that the following equality holds.

I think I may have given you the wrong advice: I'm thinking that since h > 0 (limit approached from the right) then |h| = h so you can put the h inside the absolute value sign. I'm sorry about the wrong advice.

Now one approach I am thinking of is to find a situation when you can interchange the limit signs within the absolute value from outside. I do know there are theorems that gaurantee when you can do this in some situations like this:

http://www.math.cuhk.edu.hk/course/1...onvergence.pdf