The paper gives some results and improves the derivation of the fractional Taylor's series of nondifferentiable functions obtained recently in the form fwhere E~ is the Mittag-Leffler function. Here, one defines fractional derivative as the limit of fractional difference, and by this way one can circumvent tile problem which arises with the definition of the fractional derivative of constant using Riemann-Liouville definition. As a result, a modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one's. In order to support this F-Taylor series, one shows how its first term can be obtained directly in tile form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylor's series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor's series generalizes the fractional mean value formula obtained a few years ago by Kolwantar.

a b s t r a c tIn order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann-Liouvile definition of fractional derivatives, one (Jumarie) has proposed recently an alternative referred to as a modified Riemann-Liouville definition, which directly, provides a Taylor's series of fractional order for non differentiable functions. This fractional derivative provides a fractional calculus parallel with the classical one, which applies to non-differentiable functions; and the present short article summarizes the main basic formulae so obtained.

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