# |x| and [x] differentiability

1. Dec 28, 2011

### jobsism

Are |x| and [x] differentiable anywhere? If so, what're their derivatives?

2. Dec 28, 2011

### Number Nine

Take f(x) = |x| for 0 < x < ∞. Now, what's its derivative? (you should be able to figure this out)
Is this a homework question?

3. Dec 28, 2011

### HallsofIvy

[x], in particular, (I assume you mean either the "floor" or "ceiling" function) is constant for any non-integer x.

4. Dec 28, 2011

### jobsism

I understand that the graph of |x| has got a cusp at 0, so that you can't draw a tangent there, and so itsn't differentiable at 0. I also understand that [x] (I meant the greatest integer function)isn't continuous at integral values, and so it isn't differentiable for integer values of x.

What I don't understand is the following statement made by one of my online friends: " gif is not diff at terminal points and modulus function at zero, but at other points gif gives 0 and mod gives mod x divided by x". Can anyone make sense of this? "gif" means greatest integer function.

5. Dec 28, 2011

### George Jones

Staff Emeritus
As Number Nine asked, what is the derivative of |x| for x>0? For x<0?

6. Dec 28, 2011

### mathman

The gif is constant between integers, so its derivative is 0 there. At integers it jumps, so there is no derivative.

|x| has derivative 1 for x > 0 and -1 for x < 0, this can be written as x/|x| for x ≠ 0.

7. Dec 28, 2011

### jobsism

Oh, I'm sorry I missed that question. For x>0, derivative of |x| is 1, and for x<0, derivative of |x| is -1.

Now I get it! Thanks everyone!