Some simple Questios about Derivaties

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

2. We know that continuity is defined on some closed interval , but derivative is defined on open interval . 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 , and differentiable on

(i) if for all x in , then f is increasing

(ii) if , for all x in , then f is decreasing [/TEX].

To my knowledge converse of the theorem is also true i.e if f is increasing then . In books why only way statement is written? or I am wrong.