Approximate and $L^{p}$ Peano derivatives of nonintegral order
|α|≤n fα(x)h /α! is O(h) in the approximate (resp. L) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or L 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. The function g may be chosen to be in C when u is integral, and, in any case, to have for every j of order ≤ n a bounded jth partial derivative that is Lipschitz of order u − |j|. Pointwise boundedness of order u in the L sense does not imply pointwise boundedness of the same order in the approximate sense. A classical extension theorem of Calderón and Zygmund is confirmed.