Points where a function is not differentiable

A function can be continuous at a point and still not differentiable there¹. The origin of $f(x) = |x|$ is the example.

The limit does not exist

The derivative at a point is defined as the limit of the average rate of change. At $x = 0$ the limit to examine is the following.

\lim_{h \to 0} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0} \frac{|h|}{h}

Approach	$\dfrac{\|h\|}{h}$
$h > 0$	$1$
$h < 0$	$-1$

The value differs according as $h$ approaches from the right or from the left, so the limit does not exist.

For $x > 0$ the function $|x|$ is the line $y = x$ and for $x < 0$ it is $y = -x$ . The two join at the origin, but their slopes disagree. That is why the graph comes to a point there.

Continuity and differentiability are different

Since $|x|$ joins up without a break at the origin it is continuous, but the slope is undetermined, so it is not differentiable. The converse does hold: wherever a function is differentiable it is continuous.

Candidates for a maximum or a minimum

The distinction matters when looking for a maximum or a minimum. The function $|x|$ attains its minimum $0$ at $x = 0$ , yet $f' = 0$ does not hold there.

Points where $f' = 0$
The ends of the interval
Points where the function is not differentiable

Those three kinds are the candidates for a maximum or a minimum.

Types of sharpness

Function	What happens at the origin
$\|x\|$	slopes of $1$ and $-1$ from the two sides
$x^{2/3}$	slopes of $-\infty$ and $\infty$ from the two sides
$\sqrt[3]{x}$	slope $\infty$ from both sides, a vertical tangent

The first two are pointed shapes and the last has a vertical tangent; none of them is differentiable at the origin.

Telling them apart by zooming in

Whether a function is differentiable can be judged by whether the graph looks like a line under magnification. The origin of $y = x^2$ approaches a line the more it is magnified, while the origin of $|x|$ stays a V however far one zooms in.

The shape of the derivative

Away from the origin $|x|$ is differentiable, with $f' = 1$ for $x > 0$ and $f' = -1$ for $x < 0$ . The derivative is a function with a step, matching $\operatorname{sgn} x$ , and only the origin is left undefined.

A function built with absolute values or with corners is handled by splitting the domain at the corners. Inside each piece differentiation is straightforward, and only the corners need separate treatment.

The V on the graph is $y = |x|$ , the two lines are $y = x$ and $y = -x$ , and the large dot is the non-differentiable point $(0, 0)$ .

Differentiable function, Wikipedia