When is a function not differentiable?

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SUMMARY

A function f(x) is not differentiable at specific points including cusps, non-removable discontinuities (such as jump discontinuities), and vertical slopes. Additionally, a function fails to be differentiable at locations where the limit in the definition of the derivative does not exist. These conditions are critical for understanding the behavior of functions in calculus.

PREREQUISITES
  • Understanding of calculus concepts, particularly limits and derivatives.
  • Familiarity with types of discontinuities, including removable and non-removable discontinuities.
  • Knowledge of vertical slopes and their implications on differentiability.
  • Basic grasp of function behavior and graphical analysis.
NEXT STEPS
  • Study the definition of the derivative and its implications for differentiability.
  • Explore the characteristics of different types of discontinuities in functions.
  • Learn about cusps and their effects on the differentiability of functions.
  • Investigate the concept of vertical tangents and their relation to non-differentiability.
USEFUL FOR

Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of function behavior and differentiability in mathematical analysis.

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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