**Basic Theory**

Define a time scale, or measure chain, T, to be a closed subset of the real line, R.

Define

sigma(t) = inf{s an element of T, s > t} (forward shift operator) rou(t) = sup{s an element of T, s < t} (backward shift operator)Let t be an element of T.

t is left dense if rou(t) = t,

right dense if sigma(t) = t, left scattered if rou(t) < t, right scattered if sigma(t) > t, dense if left dense or right dense.Define

Take a function f : T -> R, where R can be any Banach space, but set to the real line for simplicity.

Definition: *generalised derivative* or fdelta(t)

For every epsilon > 0 there exists a neighbourhood U of t such that

|f(sigma(t)) - f(s) - fdelta(t)(sigma(t) - s)| =< epsilon|sigma(t)-s|, for all s in U

Take T = R. Then sigma(t) = t, mu(t) = 0, fdelta = f' is the derivative used in standard calculus. If T = Z (the integers), sigma(t) = t + 1, mu(t)=1, fdelta = deltaf is the forward difference operator used in difference equations.