 One-sided derivative

ID: one-sided-derivative

The right and left one-sided derivatives are the limits of the difference quotient restricted respectively to positive and negative increments:

f_{+}^{'} (a) = lim_{h ↓ 0} \frac{f ( a + h ) - f ( a )}{h}, f_{-}^{'} (a) = lim_{h ↑ 0} \frac{f ( a + h ) - f ( a )}{h} .

At an interior point, differentiability is equivalent to existence and equality of these two finite limits. For the absolute value function,

f_{+}^{'} (0) = 1

and

f_{-}^{'} (0) = - 1

, so it is continuous at zero but not differentiable there.

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