Corner point

The first case is the one in which the two one-sided derivatives exist and are finite but different: the graph shows a “corner”.

Definition — Corner point

$x_0$ is a corner point of $f$ if the one-sided derivatives $f'_-(x_0)$, $f'_+(x_0)$ exist and are finite but different. Geometrically: the graph has two half-tangents with distinct slopes.

Example — $f(x) = |x|$ at $x_0 = 0$

For $x > 0$: $f(x) = x$, $f'(x) = 1$, so $f'_+(0) = 1$. For $x < 0$: $f(x) = -x$, $f'(x) = -1$, so $f'_-(0) = -1$. The two derivatives are $1$ and $-1$: finite and different. $x_0 = 0$ is a corner point.

At $x_0=0$ the graph of $|x|$ has a sharp angle: two half-tangents of slope $-1$ and $+1$.

Example — Piecewise function with a non-smooth join

Let $f(x) = \begin{cases} x^2 & x \le 1, \\ 3x-2 & x > 1. \end{cases}$ Continuity at $x_0 = 1$: $f(1^-) = 1$, $f(1^+) = 3\cdot 1 - 2 = 1$, $f(1) = 1$ ✓.

One-sided derivatives: $f'_-(1) = \lim_{x\to 1^-}\frac{x^2 - 1}{x-1} = \lim_{x\to 1^-}(x+1) = 2, \quad f'_+(1) = \lim_{x\to 1^+}\frac{(3x-2)-1}{x-1} = 3.$ $f'_-(1) = 2 \ne 3 = f'_+(1)$: corner point.

Links

Topics: Derivatives
Concepts: One-sided derivative · Corner point · Non-differentiability point
Methods: Non-differentiability point corner
Skills: Calculating limits · Reasoning by cases