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Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. $L^p$, 1 ≤ p ≤ ∞) sense at $x\in {ℝ}^m$ if there are numbers $f_{\alpha }\left(x\right)$, |α| ≤ n, such that $f(x+h)-{\sum }_{\left|\alpha \right|\le n}{f}_{\alpha }\left(x\right)h^{\alpha }/\alpha !$ is $O\left(h^u\right)$ in the approximate (resp. $L^p$) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or $L^p$ sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and f = g on Π....

Let $H\subset [0,1]$ be a closed set, $k$ a positive integer and $f$ a function defined on $H$ so that the $k$-th Peano derivative relative to $H$ exists. The major result of this paper is that if $H$ has finite Denjoy index, then $f$ has an extension, $F$, to $[0,1]$ which is $k$ times Peano differentiable on $[0,1]$ with $f_i=F_i$ on $H$ for $i=1,2,...,k$.

It is shown that $n$ times Peano differentiable functions defined on a closed subset of ${ℝ}^m$ and satisfying a certain condition on that set can be extended to $n$ times Peano differentiable functions defined on ${ℝ}^m$ if and only if the $n$th order Peano derivatives are Baire class one functions.