DECOMPOSITION OF I-APPROXIMATE DERIVATIVES

Abstract

It is shown that if $f : ℝ → ℝ$ has a finite $\mathcal I$-approximate derivative ${f'_{{\mathcal I} - ap}}$ everywhere in $ℝ$, then there is a sequence of perfect sets, $H_n$, whose union is $ℝ$, and a sequence of differentiable functions, $h_n$, such that $h_n = f$ over $H_n$ and ${{h'}_n} = {{f'}_{{\mathcal I} - ap}}$ over $H_n$. This result is a complete analogue of that on approximately differentiable functions by R. J. O'Malley.