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

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SUMMARY

The absolute value function |x| is not differentiable at x=0 due to the lack of a unique tangent line at that point. The limits from the left and right do not converge to the same value, indicating a discontinuity in the slope. Specifically, the left-hand limit and the right-hand limit of the derivative do not equal each other, confirming the non-differentiability. This abrupt change in the tangent line at x=0 is the definitive reason for the function's non-differentiability.

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welatiger
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Hello;

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

Thanks
 
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The limits from the left and from the right are not equal.
 
welatiger said:
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?
 
symbolipoint said:
The limits from the left and from the right are not equal.

.. What is this..

Ray Vickson said:
Just look at the graph of |x|.

Aaah, that's a physicist's answer!
 
Blast those physicists! (Especially here where that probably the best answer.)
 
Compare lim_{h \to 0^{+}} \frac{|x+h|-|x|}{h} and lim_{h \to 0^{-}} \frac{|x+h|-|x|}{h}
 
jeppetrost said:
.. What is this..



Aaah, that's a physicist's answer!

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

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