When is a function not differentiable?

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A function f(x) is not differentiable at a point if it has a cusp, a non-removable discontinuity like a jump, or an asymptote. Additionally, it is non-differentiable where the slope is vertical. The function also fails to be differentiable if it does not meet the limit criteria in the definition of the derivative. These conditions highlight the various scenarios where differentiability breaks down. Understanding these points is crucial for analyzing the behavior of functions in calculus.
Manni
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I'm curious about the conditions for when a function f(x) is not differentiable
 
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At a point, or cusp.
At a non-removable discontinuity (such as a jump discontinuity).
On an asymptote.
Probably some other places too.

[Edit]
Also at a place where the slope is vertical.
 
How about any time it fails the definition of the derivative (i.e., the limit in the definition doesn't exist).
 

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