Why Is the Absolute Value Function Not Differentiable at x=0?

[welatiger]

Hello;

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

Thanks

 

[symbolipoint]

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

 

[Ray Vickson]

Hello;

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

Thanks

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

 

[jeppetrost]

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

.. What is this..

Just look at the graph of |x|.

Aaah, that's a physicist's answer!

 

[HallsofIvy]

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

 

[Infrared]

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

 

[Ray Vickson]

.. What is this..



Aaah, that's a physicist's answer!

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

 

[symbolipoint]

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

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.