On riemann and cauchy integral on time scales

On Riemann and Cauchy integral on time scales
Camila Aversa Martins* Fernanda Alves Ozório*
Departamento de Ciências de Computação e Estatística, IBILCE, UNESP
15054-000, São José do Rio Preto, SP
E-mail: [e-mail]

Luciano Barbanti
Departamento de Matemática, FEIS, UNESP
15385-000, Ilha Solteira, SP
E-mail: [e-mail]

ABSTRACT

The Riemann integral

∫ f (s)ds of a function

f : R → R , can be defined in several

ways as for instance by using the Darboux approach. We use here the definition of the
Riemann integral on general time scales T (that is, T is nonempty and closed subset of R) done by G. Guseinov. After we define a Cauchy-type integral that follows the one defined by Price in the case T= R and will be presenting results on the class of the function f for wich the two concepts are coincident.
Keyword: time scales, Riemann integral, Cauchy integral.
Here we will be working on the frame of the Calculus in time scales T (that is a closed subset of R) introduced by S. Hilger in 1988 and will be following the standard concepts and notations presented in [1].
In the work by G. Guseinov [2] it is presented the definition of the Riemann integral for a bounded real function.
Let be the class P [a ,b ]T of all partitions of I = [a, b]T .
Definition 1: Consider a bounded function on [a, b[T and let P = a = t 0 < t 1 < ... < tn = b be a partition of [a, b[T . In each interval [ti − 1, ti[ , where 1 ≤ i ≤ n choose an arbitrary point form the sum S =

ξi

and

n

∑ f (ξ )(t − t i i

).

i −1

i =1

We call S a Riemann sum of f corresponding to the partition P. We say that f is
Riemann integrable from a to b, if there exist a number I with the following property.
For each ε > 0 , there exist δ > 0 such that S − I < ε for every Riemann sum S of f corresponding to a partition P ∈ Pε (see the definition 2.8 in [2]) independently of the way in

which we choose ξi ∈ [ti −1,ti ] , 1 ≤ i ≤