Second Derivative - Limit

Limit

It is possible to write a single limit for the second derivative:

The expression on the right can be written as a difference quotient of difference quotients:

This limit can be viewed as a continuous version of the second difference for sequences.

Please note that the existence of the above limit does not mean that the function has a second derivative. The limit above just give a possibility for calculating the second derivative but does not provide a definition. As a counterexample look on the sign function which is defined through

\sgn(x) = \begin{cases} -1 & \text{if } x < 0, \\ 0 & \text{if } x = 0, \\ 1 & \text{if } x > 0. \end{cases}

The sign function is not continuous at null and therefore the second derivative for does not exist. But the above limit exists for :

\begin{align} \lim_{h \to 0} \frac{\sgn(0+h) - 2\sgn(0) + \sgn(0-h)}{h^2} &= \lim_{h \to 0} \frac{1 - 2\cdot 0 + (-1)}{h^2} \\ &= \lim_{h \to 0} \frac{0}{h^2} \\ &= 0 \end{align}

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