Analysis manuscript: small area error, different limits
Develop and revise one connected manuscript about two ways of measuring error between continuous functions. Construct a Cauchy sequence that loses continuity in integral distance, prove that uniform distance prevents that failure, and identify which measurements remain stable. Keep exact calculations, rejected arguments, theorem dependencies and actual feedback. This public, supported portfolio is not an unseen examination or automatic certification of proof correctness.
Planning estimate: 14 hours, including analysis, writing and revision. Use the investigation and working agreement below. Keep a single coherent report across milestones rather than unrelated short answers.
Preparation and working agreement
Work with real functions on I=[−1,1] and ordinary Riemann integrals. Let C(I) mean the set of continuous real functions on I. Define d₁(f,g)=∫I |f−g| and d∞(f,g)=sup over x∈I of |f(x)−g(x)|. A metric is a finite nonnegative symmetric distance, zero exactly for identical functions, satisfying the triangle inequality. A sequence is Cauchy when every pair of sufficiently late terms is arbitrarily close in the chosen distance; completeness means every Cauchy sequence has a limit inside the space.
You may use completeness of the real numbers, the extreme value theorem, continuity of sums/products/absolute values, and the established Riemann integral properties: linearity, order, interval additivity, and integrability of continuous functions and step functions. State these dependencies. Prove any further convergence statement used here rather than assuming a measure-theory theorem or an interchange of limit and integral.
Define H(x)=0 for −1≤x≤0 and H(x)=1 for 0<x≤1. For each integer n≥1, define fₙ(x)=0 for x≤0, fₙ(x)=nx for 0≤x≤1/n, and fₙ(x)=1 for x≥1/n. The overlapping formulas agree at their junctions. H is a bounded integrable comparison function, but is not in C(I). Distinguish a comparison integral from a distance whose two arguments are required to lie in C(I).
Sketch the cases n=1,2,4 on common axes, label the ramp widths and triangular discrepancies, and prepare a table of exact integrals. A graph or numerical quadrature is exploratory evidence, not a proof about every n or every possible limit.
Retain a dated source/assistance ledger, a first conjecture, an initial proof, specific criticism, and a revised manuscript. Suggested hours are an author estimate, not measured learner time. Public solutions count as prior exposure. If no peer or teacher actually reviews the work, leave independent review pending and label any defense as self-review.
Use each argument's planning fields, immutable revisions and revision-specific feedback. Save derivations, data provenance, numerical checks and a reproducible method in the report. External code or files should be identified by filename and instructions in your response; this page stores text and does not run code or upload artifacts.
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0 of 6 milestones have recorded responses. This counts submissions, not quality, independent review or project mastery.
Define the two spaces of error and prove their contracts
Prove that d₁ and d∞ are metrics on C(I). In particular, prove that a continuous nonnegative function with integral zero vanishes everywhere, including at an endpoint. Derive d₁(f,g)≤2d∞(f,g). Audit the proposed enlargement to all functions whose absolute value is Riemann integrable: is ∫|f−g| even defined for every pair? Give a counterexample. State exactly which domain your manuscript will use.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compute the sequence and certify Cauchy behavior
Check every junction in fₙ and compute ∫I fₙ and ∫I |fₙ−H|. For m≥n, determine the pointwise order of fₘ and fₙ and obtain an exact formula for d₁(fₙ,fₘ). Give an explicit Cauchy cutoff for a prescribed ε>0. Determine the pointwise limit, then prove this sequence is not Cauchy in d∞ using pairs of arbitrarily late indices. Record the n=1,2,4 table and explain what its finite evidence does not prove.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Rule out every continuous integral-distance limit
The claim “the pointwise limit is discontinuous, so no d₁ limit exists in C(I)” skips a step: d₁ convergence need not imply pointwise convergence. Repair it. Assume an arbitrary g∈C(I) satisfies d₁(fₙ,g)→0. Use H only as an integrable comparison function to prove ∫I |g−H|=0. Deduce g(x)=0 for x<0 and g(x)=1 for x>0 using local interval positivity. Finish the contradiction at 0 and state precisely which metric space is incomplete.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove completeness in the stronger distance
Prove directly that (C(I),d∞) is complete. Starting from a d∞-Cauchy sequence uₙ, construct its pointwise candidate using scalar completeness. Keep one cutoff valid for all inputs when passing to the limit in a second index. Prove the candidate continuous by a three-error argument before asserting it belongs to C(I). Explain why d₁≤2d∞ transfers convergence in only one direction, and use the ramp sequence to refute the reverse Cauchy implication.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Separate stable measurements from unstable readings
For a fixed continuous weight w on I define M_w(f)=∫I w(x)f(x) dx. Prove |M_w(f)−M_w(g)|≤||w||∞ d₁(f,g). Apply it to the ramps and H, and compute the exact error for w(x)=x. Next construct continuous functions zₙ that approach 0 in integral distance but have zₙ(0)=1. Use them to disprove stability of point evaluation and of the maximum value in integral distance. Compare point evaluation in uniform distance. State what these results do and do not say about measuring an unknown function.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Revise and defend one complete manuscript
Assemble the definitions, metric proofs, ramp calculation, failure of integral-distance completeness, uniform completeness and measurement theorem into one manuscript. Audit these shortcuts: “Cauchy means convergent in any space”; “a discontinuous pointwise limit alone rules out an integral-distance limit”; “zero integral forces every bounded nonnegative function to vanish”; “a small integral error controls each reading.” For each, supply the missing hypothesis or a counterexample. Preserve an earlier draft and a change log. Ask a reviewer to challenge one quantified step and one changed example; record the actual response and reviewed revision. If none is available, conduct a self-defense and leave independent review pending.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review, defend and revise
Ask a reviewer to examine the complete reasoning, reproduce selected results and pose an unfamiliar case. Record their feedback against the specific revision discussed. Reviewer labels are entered locally and do not verify identity or judgment. A complete set of text submissions does not certify an independently defended investigation.
Targeted lesson returns
Define the two spaces of error and prove their contracts
