prove greatest integer function is not differentiable

[PaulKraemer]

Hi,

Can anyone help me prove that if n is any integer, then the greatest integer function f is not differentiable at n.

This problem is in the chapter where derivatives are calculated using the following limit:

f'(a) = lim as h->0 of [f(a+h) - f(a)] / h

There is an example in this chapter where they prove that f(x) = |x| is not differentiable at zero. In this example, they use the fact that the left hand and right hand derivatives of f at zero are not equal.

For the greatest integer function, it is obvious that this function is not continuous at every integer n, but it seems to me like whether you approach an integer n from the left or the right, the derivative (slope) would always be zero.

If anyone can help me understand this, I'd really appreciate it.

Thanks in advance,
Paul

 

[JeffM]

Hi,

Can anyone help me prove that if n is any integer, then the greatest integer function f is not differentiable at n.

This problem is in the chapter where derivatives are calculated using the following limit:

f'(a) = lim as h->0 of [f(a+h) - f(a)] / h

There is an example in this chapter where they prove that f(x) = |x| is not differentiable at zero. In this example, they use the fact that the left hand and right hand derivatives of f at zero are not equal.

For the greatest integer function, it is obvious that this function is not continuous at every integer n, but it seems to me like whether you approach an integer n from the left or the right, the derivative (slope) would always be zero.

If anyone can help me understand this, I'd really appreciate it.

Thanks in advance,
Paul

I'll take a go.

One answer is that differentiability entails continuity. As you yourself point out, the greatest integer function is not continuous at any integer n so it is not differentiable. I must admit that this answer does not speak to one's intuition.

A different answer arises out of the definition of the derivative. f'(a) = limit as h --> 0 of {[f(a + h) - f(a)] / h}. If f(x) is the greatest integer function and n is a non-negative integer, the limit as h approaches 0 from the right is {[f(n + h) - f(n)] / h} = [(n - n) / h] = (0 / h) = 0. The limit as h approaches from the left is
{f(n + h) - f(n)] / h} = {[(n - 1) - n] / h} = - 1/ h, which does not equal zero and in fact increases without bound. So there is no limit and therefore no derivative.

 

[galactus]

If I may. I believe this is what Jeff is trying to say:

Let \(\displaystyle [x]\) denote the Greatest Integer Function.

If a function is not continuous at x=c, then it is not differentiable at x=c.

Note:

From the left:

\(\displaystyle \lim_{x\to 0^{-}}\frac{f(x)-f(0)}{x-0}=\lim_{x\to 0^{-}}\frac{[x]-0}{x}=\infty\)

From the right:

\(\displaystyle \lim_{x\to 0^{+}}\frac{f(x)-f(0)}{x-0}=\lim_{x\to 0^{+}}\frac{[x]-0}{x}=0\)

As Jeff mentioned, two different limits when approaching from different directions.

Although it is true that differentiability implies continuity, the converse is not true.

That is, it is possible for a function to be continuous but not differentiable at x=c.

But, not the other way around.

Did you know there is a function, or functions, out there that are continuous everywhere, but differentiable nowhere.

 

[JeffM]

If I may. I believe this is what Jeff is trying to say: You expressed exactly what I fumbling to say. Thanks, galactus

 

[PaulKraemer]

Thank you JeffM and Galactus -

That was a big help!

Paul