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When the Derivative Does Not Exist

Not every continuous function is differentiable. The derivative fails to exist when the limit $\lim_{h \to 0} \frac{f(x_0+h) - f(x_0)}{h}$ does not exist. There are three classical failure modes: kinks (corners), vertical tangents, and wild oscillations.

A kink occurs when the left-hand and right-hand derivatives differ: the function approaches a corner from different angles. Classic example: $f(x) = |x|$ at $x_0 = 0$ — the left-derivative is $-1$ and the right-derivative is $+1$ . A vertical tangent occurs when the limit of the difference quotient is $\pm\infty$ — the graph becomes vertical. Wild oscillations like $f(x) = x \sin(1/x)$ near $x = 0$ produce a limit that does not exist.

Formal View

Example 10.1 — Non-Differentiable at a Kink

The function

f(x) = |x|

satisfies

f'(x) = -1

for

x < 0

and

f'(x) = +1

for

x > 0

. At

x = 0

\lim_{h\to 0^-} \frac{|h|}{h} = -1 \neq +1 = \lim_{h\to 0^+} \frac{|h|}{h}

. The two-sided limit does not exist, so

f

is not differentiable at

0

Even though $|x|$ is continuous everywhere, it is NOT differentiable at $x=0$ . Continuity does not imply differentiability.

Why This Matters

Knowing when derivatives fail is essential in optimization — non-smooth functions require specialized algorithms.

ReLU activation $\max(0,x)$ has a kink at $x = 0$ — it is not differentiable there, but subgradients are used in practice.
L1 regularization $|w|$ creates kinks in the loss landscape — solved by proximal methods or subgradient descent.
Contact mechanics: physical contact creates kinks in displacement fields.

Learning Resources

Non-differentiable functions

Khan Academy

Examples of continuous functions that are not differentiable.

7 min

Differentiability and continuity

Khan Academy

Why differentiability implies continuity, but not vice versa.

8 min

Quiz

Question 1

At $x = 0$ , $f(x) = |x|$ is:

Question 2

If a function is differentiable at $x_0$ , it must also be continuous at $x_0$ .

Common Mistakes

Thinking continuity implies differentiability — the function $|x|$ is the standard counterexample.
Forgetting that a vertical tangent also means non-differentiability (limit is $\pm\infty$ , not a finite number).