Editing Time-scale calculus (section)

==Derivative==
Take a function:
:<math>f: \mathbb{T} \to \R,</math>

(where '''R''' could be any [[Banach space]], but is set to  the real line for simplicity).

Definition: The ''delta derivative'' (also Hilger derivative) <math>f^{\Delta}(t)</math> exists if and only if:

For every <math>\varepsilon > 0</math> there exists a neighborhood <math>U</math> of <math>t</math> such that:
:<math>\left|f(\sigma(t))-f(s)- f^{\Delta}(t)(\sigma(t)-s)\right| \le \varepsilon\left|\sigma(t)-s\right|</math>
for all <math>s</math> in <math>U</math>.

Take <math>\mathbb{T} =\mathbb{R}.</math>  Then <math>\sigma(t) = t</math>, <math>\mu(t) = 0</math>, <math>f^{\Delta} = f'</math>; is the derivative used in standard [[calculus]]. If <math>\mathbb{T} = \mathbb{Z}</math> (the [[integer]]s), <math>\sigma(t)  = t + 1</math>, <math>\mu(t)=1</math>, <math>f^{\Delta} = \Delta f</math> is the [[forward difference operator]] used in difference equations.