# The differenation of abs(x)

1. Aug 27, 2013

### welatiger

Hello;

I want to know why |x| is not differentiable at x=0.

Thanks

2. Aug 27, 2013

### symbolipoint

The limits from the left and from the right are not equal.

3. Aug 27, 2013

### Ray Vickson

Just look at the graph of |x|. Does it have a unique tangent at x=0?

4. Aug 27, 2013

### jeppetrost

.. What is this..

5. Aug 27, 2013

### HallsofIvy

Staff Emeritus
Blast those physicists! (Especially here where that probably the best answer.)

6. Aug 27, 2013

### Infrared

Compare $$lim_{h \to 0^{+}} \frac{|x+h|-|x|}{h}$$ and $$lim_{h \to 0^{-}} \frac{|x+h|-|x|}{h}$$

7. Aug 27, 2013

### Ray Vickson

More "mathematical": the subgradient is not a singleton at x = 0.

8. Aug 27, 2013

### symbolipoint

I should not have said that; it is incorrect.

Actually, the change in tangent line makes an abrupt, not continuous change at x=0. The lack of a continuous change is the reason why the absolute value of x is not differentiable there.