Differentiability and Non-Differentiable Points
Identify conditions under which a function is not differentiable (corners, cusps, vertical tangents, discontinuities). Prove that differentiability implies continuity. Connect to AP exam graph analysis questions.
A lesson of AP Calculus AB, in Differentiation: Definition and Fundamental Properties, about 10 minutes.
Differentiability and the Relationship to Continuity
By the end of this lesson, you will be able to determine where a function is differentiable, explain why differentiability implies continuity (but not vice versa), and identify points of non-differentiability.
The Derivative as a Limit
Recall: $$f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$$
A function is differentiable at $x = a$ if this limit exists (is a finite number).
Differentiability Implies Continuity
Theorem: If $f$ is differentiable at $x = a$, then $f$ is continuous at $x = a$.
Proof sketch: If $f'(a)$ exists, then $\lim_{h\to 0}[f(a+h)-f(a)] = f'(a) \cdot 0 = 0$, which means $\lim_{x\to a} f(x) = f(a)$ — the definition of continuity.
The converse is false. Continuity does NOT guarantee differentiability. The canonical example: $f(x) = |x|$ is continuous everywhere but not differentiable at $x = 0$.
Where Differentiability Fails
A function fails to be differentiable at a point when the limit defining the derivative doesn't exist. This happens at:
1. Corners (sharp turns)
$f(x) = |x|$ has a corner at $x = 0$.
Left derivative: $\lim_{h\to 0^-}\frac{|h|-0}{h} = \lim_{h\to 0^-}\frac{-h}{h} = -1$
Right derivative: $\lim_{h\to 0^+}\frac{h}{h} = 1$
The one-sided derivatives exist but are unequal → the two-sided limit DNE → not differentiable at $x = 0$.
2. Cusps
A cusp is a "pinched" corner where the function turns back sharply. Example: $f(x) = x^{2/3}$ at $x = 0$. The derivative approaches $\pm\infty$ from both sides.
3. Vertical tangent lines
If $f'(a) = \pm\infty$, the tangent line is vertical and the derivative is undefined.
4. Discontinuities
If $f$ is discontinuous at $x = a$, it cannot be differentiable there (since differentiability implies continuity).
Left and Right Derivatives
Left-hand derivative: $f'-(a) = \lim{h\to 0^-}\frac{f(a+h)-f(a)}{h}$
Right-hand derivative: $f'+(a) = \lim{h\to 0^+}\frac{f(a+h)-f(a)}{h}$
$f$ is differentiable at $a$ ↔ $f'+(a)$ (both exist and are equal).