Absolute Value Function |x|: Graph, Derivative, Integral and Corner

The absolute value function f(x) = |x| gives the distance from x to 0 on the number line, so it removes any negative sign: |3| = 3 and |-3| = 3. Its V-shaped graph is the classic example of a function that is continuous everywhere but has a point where the derivative does not exist.

Formula

It is a piecewise function built from two linear functions. Equivalently, |x| = \sqrt{x^2}, which links it to the square root function.

Graph of |x|

The general form y = a|x - h| + k is a V with vertex (h, k), opening up if a > 0 and down if a < 0, with side slopes \pm a.

Key properties

• Domain all reals; range [0, \infty).
• Even function: |-x| = |x|, symmetric about the y-axis.
• Continuous everywhere, including at 0.
• Absolute minimum of 0 at x = 0, which is a critical point because f'(0) does not exist.
• Triangle inequality: |a + b| \le |a| + |b|.
• Products split, sums do not: |ab| = |a||b|, but |a + b| is usually not |a| + |b|.

Derivative of |x|

For x \ne 0:

At x = 0 the one-sided limits of the difference quotient are

They disagree, so f'(0) does not exist. This is the textbook case of when derivatives fail: a corner. With the chain rule, \frac{d}{dx}|g(x)| = \frac{g(x)}{|g(x)|}g'(x) wherever g(x) \ne 0.

Integral of |x|

Check it piece by piece: for x > 0 it is \frac{x^2}{2}, with derivative x = |x|; for x < 0 it is -\frac{x^2}{2}, with derivative -x = |x|. For a definite integral, split at the point where the inside changes sign.

Worked examples

Example 1: definite integral. Evaluate \int_{-1}^{2}|x|\,dx by splitting at 0:

Geometrically, these are two triangles with areas \frac{1}{2} and 2.

Example 2: a shifted absolute value. g(x) = |x - 3| has derivative 1 for x > 3 and -1 for x < 3, and is not differentiable at x = 3.

Example 3: chain rule. For x \ne 2, \frac{d}{dx}|2x - 4| = \frac{2x - 4}{|2x - 4|} \cdot 2, which is 2 for x > 2 and -2 for x < 2.

Example 4: a limit. \lim_{x \to 0}\frac{|x|}{x} does not exist, because it is 1 from the right and -1 from the left.

Common mistakes

1. Saying |x| is not continuous at 0. It is continuous; it is only not differentiable there.
2. Writing |{-x}| = -|x|. Absolute value is never negative: |-x| = |x|.
3. Integrating across 0 without splitting. \int_{-1}^{2}|x|\,dx is not \frac{2^2}{2} - \frac{(-1)^2}{2}.
4. Forgetting it is a critical point. When looking for extrema, include points where the derivative does not exist, like the vertex of a V.

How it connects

Absolute values appear in \int \frac{1}{x}\,dx = \ln|x| + C, in the epsilon-delta definition of a limit, and in absolute convergence of series in convergence tests. For more on corners and other places where derivatives fail, see the continuity guide.