Mathematics Weblog

Generalisation of derivative

Sunday 6 June 2004 at 2:25 pm | In Articles | Post Comment

Inspired by a posting on S.O.S. Mathematics CyberBoard

Most students will be familiar with the definition of the derivative of a real-valued function of a real variable defined on some interval (a,b):

$x \in (a,b)$

$\displaystyle \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$

$f^{\prime}(x)$

It is also clear that for this to make sense must be defined at (and of course it is a well-known consequence of the definition that is also continuous at ). But what if is defined on (a,b) but not at , can we do anything then? Yes, we can define a pseudo-derivative $\Lambda f$ of provided is defined on a neighbourhood of :

$\displaystyle \Lambda f(x)=\lim_{h \to 0}\frac{f(x+h)-f(x-h)}{2h}$

This pseudo-derivative has similar properties to the derivative and indeed it has the same values where is differentiable but there are significant differences as the following exercises show:

If is differentiable at show that $\Lambda f(x)=f^{\prime}x)$

If show that $\Lambda f(0)$ exists although $f^{\prime}(0)$ does not

If $f(x)=\left\{\begin{array}{ll}x & \mbox{ if } x<0\\-x^2 & \mbox{ if } x \geq 0\end{array}\right.$ show that has a local maximum at 0 but $\Lambda f(0) \neq 0$

Suppose is differentiable on , except at a point in , with $f^{\prime}(x)\geq 0$ for $x \neq x_0$ .
If $\Lambda (f)(x_0)$ exists show that $\Lambda (f)(x)\geq0$