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		of which is correct), the actual value of f'(0) can be determined in the limit. This is possible because

		Thus  p 1(f'(0)) can be obtained from G' (and hence G) in the limit. Also, it is easy to determine f'(0), in the limit, from  p 1(f'(0)) and from f'(0), a  j -program for f' can easily be found.

		8.18 Corollary .

		We leave it to the reader to modify the above proof to show the following result.

		8.19 Proposition For each , .

		8.20 Corollary For each , .

		Propositions 8.17 and 8.19 may be paraphrased this way: there are collections of functions that can be learned from the most offensive noisy texts (*-noisy texts), but not from the least offensive incomplete texts (1-incomplete texts). It is natural to ask whether the reverse is true, i.e., whether there are collections of functions that can be learned from the most offensive incomplete texts (*-incomplete texts), but not from the least offensive noisy texts (1-noisy texts). Proposition 8.21 below answers this question negatively, thereby implying that—in the context of function identification—missing data are strictly more harmful than noisy data.

		8.21 Proposition Let a,. Then:

		(a) .

		(b) .

		Proof: Let scientist M be given. We construct a scientist M' such that and .

		Let be such that, for all  s , all , and all x,,