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A new calculus based on fractal subsets of the real line is formulated. Inthis calculus, an integral of order $\alpha, 0 < \alpha \leq 1$, called$F^\alpha$-integral, is defined, which is suitable to integrate functions withfractal support $F$ of dimension $\alpha$. Further, a derivative of order$\alpha, 0 < \alpha \leq 1$, called $F^\alpha$-derivative, is defined, whichenables us to differentiate functions, like the Cantor staircase, ``changing''only on a fractal set. The $F^\alpha$-derivative is local unlike the classicalfractional derivative. The $F^\alpha$-calculus retains much of the simplicityof ordinary calculus. Several results including analogues of fundamentaltheorems of calculus are proved.  The integral staircase function, which is a generalisation of the functionslike the Cantor staircase function, plays a key role in this formulation.Further, it gives rise to a new definition of dimension, the$\gamma$-dimension.  $F^\alpha$-differential equations are equations involving$F^\alpha$-derivatives. They can be used to model sublinear dynamical systemsand fractal time processes, since sublinear behaviours are associated withstaircase-like functions which occur naturally as their solutions. As examples,we discuss a fractal-time diffusion equation, and one dimensional motion of aparticle undergoing friction in a fractal medium.

CALCULUS ON FRACTAL SUBSETS OF REAL LINE — I: FORMULATION