sk of E over each Bk such that jksk(xk)=0 but |Dsk(xk)|≥C>0. Let ρ(x) denote a standard bump function for the unit ball at the origin: a smooth real-valued function which is equal to 1 on B1/2(0), which vanishes to infinite order on the boundary of the unit ball. Consider every other section s2k. At x2k, these satisfy j2ks2k(x2k)=0. Suppose that 2k is given. Then, since these functions are smooth and each satisfy j2k(s2k)(x2k)=0, it is possible to specify a smaller ball B′δ(x2k) such that the higher order derivatives obey the following estimate: where Now is a standard bump function supported in B′δ(x2k), and the derivative of the product s2kρ2k is bounded in such a way that As a result, because the following series and all of the partial sums of its derivatives converge uniformly q(y) is a smooth function on all of V. We now observe that since s2k and 2ks2k are equal in a neighborhood of x2k, So by continuity |Dq(x)|≥ C>0. On the other hand, since Dq(x2k+1)=0 because q is identically zero in B2k+1 and D is support non-increasing. So Dq(x)=0. This is a contradiction. We now prove Lemma 2. First, let us dispense with the constant C from the first lemma. We show that, under the same hypotheses as Lemma 1, |Ds(y)|=0. Pick a y in V\{x} so that jks(y)=0 but |Ds(y)|=g>0. Rescale s by a factor of 2C/g. Then if g is non-zero, by the linearity of D, |Ds(y)|=2C>C, which is impossible by Lemma 1. This proves the theorem in the punctured neighborhood V\{x}. Now, we must continue the differential operator to the central point x in the punctured neighborhood. D is a linear differential operator with smooth coefficients. Furthermore, it sends germs of smooth functions to germs of smooth functions at x as well. Thus the coefficients of D are also smooth at x. A specialized application Let M be a compact smooth manifold (possibly with boundary), and E and F be finite dimensional vector bundles on M. Let be the collection of smooth sections of E. An operator is a smooth function (of Fréchet manifolds) which is linear on the fibres and respects the base point on M: The Peetre theorem asserts that for each operator D, there exists an integer k such that D is a differential operator of order k. Specifically, we can decompose where is a mapping from the jets of