# Thread: Lower and upper derivaties of function at point

1. ## Lower and upper derivaties of function at point

Hi guys,
I'm looking for a little help with computing the lower and upper derivatives of my function given below, at a point $\displaystyle x\in (a, b)$.

$\displaystyle f(x) = \sum_{k=1}^{\infty}\left [ {length( (c_{k}, d_{k})\cap (-\infty, x) )} \right ]$

(where $\displaystyle \left \{ (c_{k}, d_{k})_{k=1}^{\infty} \right \}$ is a countable collection of open intervals in (a, b) and $\displaystyle \sum_{k=1}^{\infty}(d_{k} - c_{k}) < \infty$)

The lower and upper derivatives of a real-value function are defined as such:
$\displaystyle Uf(x) = \lim_{h \rightarrow 0} \left[\sup_{t\in (0, h]}{\frac{f(x+t) - f(x)}{t}}\right]$

$\displaystyle Lf(x) = \lim_{h \rightarrow 0} \left[\inf_{t\in (0, h]}{\frac{f(x+t) - f(x)}{t}}\right]$

This is what I have, could someone tell me if this is wrong? I have some doubts.

If $\displaystyle f(x+t)= \sum_{k=1}^{\infty} \left[ min(d_{k}, x+t) - c_{k}\right ]$

$\displaystyle Uf(x) = \lim_{h \rightarrow 0} \left[\sup_{t\in (0, h]}{\frac{ \sum_{k=1}^{\infty} \left[ (min(d_{k}, x+t) - c_{k}) - (min(d_{k}, x) - c_{k}) \right ]}{t}}\right]$

$\displaystyle = \lim_{h \rightarrow 0} \left[\sup_{t\in (0, h]}{\frac{ \sum_{k=1}^{\infty} \left[ min(d_{k}, x+t) - min(d_{k}, x) \right ]}{t}}\right] = \lim_{h \rightarrow 0} \left[\sup_{t\in (0, h]}{\frac{ \sum_{k=1}^{\infty} \left \{ 0 \leqslant y \leqslant t \right \} }{t}}\right] = \infty$

Similarly,

$\displaystyle Lf(x)= \lim_{h \rightarrow 0} \left[\inf_{t\in (0, h]}{\frac{ \sum_{k=1}^{\infty} \left[ min(d_{k}, x+t) - min(d_{k}, x) \right ]}{t}}\right] = \lim_{h \rightarrow 0} \left[\inf_{t\in (0, h]}{\frac{ \sum_{k=1}^{\infty} \left \{ 0 \leqslant y \leqslant t \right \} }{t}}\right] = 0$