Measure theory 1.1.3: require boundedness in Exercise 1.1.23 - #631
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Exercise 1.1.23 asks to show that any continuous function on [a,b] is
Riemann integrable, and then "more generally, that any *bounded*,
piecewise continuous function f: [a,b] -> R is Riemann integrable".
RiemannIntegrableOn.piecewise_continuous dropped the boundedness
hypothesis, which makes it false: partition [0,1] into {0} and (0,1],
and take f x = 1/x (so f 0 = 0 by the junk-value convention). Then f is
continuous on each piece, but unbounded, and unbounded functions are
never Riemann integrable -- as this file already records in
RiemannIntegrable.bounded.
Also gives the lemma the Exercise 1.1.23' docstring it was missing.
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
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Checked against An Introduction to Measure Theory, Exercise 1.1.23 (§1.1.3, p. 16):
(with footnote 8: "A function
f: [a,b] → Ris piecewise continuous if one can partition[a,b]into finitely many intervals, such thatfis continuous on each interval".)RiemannIntegrableOn.piecewise_continuousomits the boundedness hypothesis, which makes the statement false.Counterexample. Take
I = Icc 0 1, partitioned intoIcc 0 0andIoc 0 1, andf = fun x ↦ 1/x(sof 0 = 0by the junk-value convention).fisContinuousOneach piece — trivially on the singleton, and1/xis continuous on(0,1]— sohcontholds. Butfis unbounded on[0,1], and unbounded functions are never Riemann integrable; the text notes this right after Definition 1.1.5, and this file already records it asRiemannIntegrable.bounded. So thesorryis unprovable as stated.The added hypothesis uses the same idiom as
RiemannIntegrable.boundedin this file, so the two compose directly. No call sites are affected — the lemma is not yet used anywhere.Also adds the
Exercise 1.1.23'docstring, which this lemma was missing (theExercise 1.1.23label sits onRiemannIntegrableOn.continuousjust above; the text's exercise has these two sentences and no lettered parts).