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Let $fH\subset ℝ\to ℝ$ be $k$ times differentiable in both the usual (iterative) and Peano senses. We investigate when the usual derivatives and the corresponding Peano derivatives are different and the nature of the set where they are different.

For subspaces, $X$ and $Y$, of the space, $D$, of all derivatives $M(X,Y)$ denotes the set of all $g\in D$ such that $fg\in Y$ for all $f\in X$. Subspaces of $D$ are defined depending on a parameter $p\in [0,\infty ]$. In Section 6, $M(X,D)$ is determined for each of these subspaces and in Section 7, $M(X,Y)$ is found for $X$ and $Y$ any of these subspaces. In Section 3, $M(X,D)$ is determined for other spaces of functions on $[0,1]$ related to continuity and higher order differentiation.

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.