One-sided derivatives

So far we have worked with differentiable functions. But there exist functions which, although continuous, at certain points admit no derivative: the tangent to the graph does not exist as a unique line, or it exists but is vertical (hence not representable as $y=mx+q$). These are the non-differentiability points. To study them, one splits the limit of the difference quotient into its two one-sided limits.

Definition — Right and left derivative

The right derivative and the left derivative of $f$ at $x_0$ are the one-sided limits of the difference quotient: $f'_+(x_0) = \lim_{\Delta x\to 0^+}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}, \qquad f'_-(x_0) = \lim_{\Delta x\to 0^-}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}.$ The derivative $f'(x_0)$ exists and is finite if and only if $f'_+(x_0) = f'_-(x_0)$ and both are finite.

Remark — " $=$" between two limits: a precise reading

The condition "$f'_+(x_0) = f'_-(x_0)$" is an equation between two limits, not between two “numbers” we already have: to write it truthfully both limits must first exist (finite). Esty observes that the sign "$=$" between objects defined as limits means two things: (1) both sides are well defined (they exist as reals); (2) the values coincide. In the cusp / vertical tangent cases one of the two points already fails at step (1) — and that is why the derivative, in the classical sense, does not exist.

Links

Topics: Derivatives
Concepts: One-sided derivative · Limit · Non-differentiability point
Skills: Calculating limits