Dear All,

I have a simple question which creates confusion for me

1. What is derivative of zero function? I know derivative of constant is zero and derivative of ZERO is zero. But the problem is derivative of ZERO Function at Zero becomes indeterminate form i.e

$\displaystyle f'(x)=\lim_{x \to \0} \frac{\mathrm{0-0} }{\mathrm{x-0} } $

2. We know that continuity is defined on some closed interval $\displaystyle [a, b]$, but derivative is defined on open interval $\displaystyle (a, b)$. Why?

3. In almost every book of calculus, theorem about test of increasing and decreasing function is given one way as:

If f is continuous on closed interval $\displaystyle [a, b]$, and differentiable on $\displaystyle (a, b),$

(i) if $\displaystyle f'(x)>0, $ for all x in$\displaystyle (a, b)$, then f is increasing

(ii) if $\displaystyle f'(x)<0 $, for all x in $\displaystyle (a, b)$, then f is decreasing [/TEX].

To my knowledge converse of the theorem is also true i.e if f is increasing then $\displaystyle f'(x)>0, $. In books why only way statement is written? or I am wrong.