AP Calculus BC / Unit 2: Differentiation: Definition and Fundamental Properties
2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
Notes
- If $f$ is differentiable at $a$, then $f$ is continuous at $a$.
- The reverse is false: continuous functions can fail to be differentiable.
- A derivative fails to exist at a corner, a cusp, a vertical tangent, or any discontinuity.
Practice
Where is $f(x)=|x-2|$ not differentiable?
It has a corner at $x=2$.
If $f$ is not continuous at $x=1$, then:
Differentiability requires continuity.
$f(x)=x^{1/3}$ is not differentiable at $x=0$ because:
$\displaystyle f'(x)=\frac{1}{3}x^{-2/3}$ blows up at 0.
Ready for more? Take the Unit 2 test.