{"id":"fd8ff50e-b09c-43c4-9315-a06b78ff99e6","arxiv_id":"2009.13719","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"There exist continuous functions on [0,1] for which the limit of y_n = x_{n+1} - x_n does not exist, answering Furdui's open Problem 1.32(b) in the negative.","lead":"A math paper settles an open question about Riemann sums for continuous functions. It shows that the differences of successive Riemann sums can fail to converge even for a continuous function, using a probabilistic proof with Brownian motion.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The paper's central claim is an existence theorem, proved by contradiction from a probabilistic lemma. I checked the probabilistic input: Brownian motion has continuous sample paths (standard; the paper cites Mörters and Peres), and the event that y_n converges is measurable. The contradiction argument is valid: assuming Theorem 3 is false makes the convergence set have probability 1 for f=W, forcing the gap y_{4s}-y_{2s} to 0 in distribution, while Lemma 4 gives a nondegenerate normal limit. The variance computations are the real heart of the paper, and I found them correct after independent re-derivation. The only errant displayed object is the upper limit 4s-1 in one inner sum in Lemma 4; the printed value is consistent with the correct upper limit 4s, so it is a harmless typo. Propositions 1 and 2 are standard and correctly proved; the AC proof's Riemann-sum/indicator step and the Stolz-Cesaro application both check out. The author's admission that the proof is non-constructive and the explicit construction remains open is a limitation, not a defect. I therefore see no reason to change the reader's ACCEPT verdict.","tokens_in":3791,"tokens_out":28506,"duration_ms":220903,"concrete_test":"Independently recompute the four covariance identities in Lemma 4 by symbolic summation (or by numerical evaluation for s=1,...,10): verify (i) E x_{4s}x_{2s}=(32s^2-18s+1)/12, (ii) E x_{4s+1}x_{2s+1}=(32s^3+14s^2)/(12s+3), (iii) E x_{4s+1}x_{2s}=(32s^3-2s^2-5s-1)/(12s+3), (iv) E x_{4s}x_{2s+1}=(64s^3+28s^2-7s-1)/(24s+12), and then verify E(y_{4s}-y_{2s})^2 = 1/2+1/2-2E y_{4s}y_{2s} -> 1/4. If the limit is not 1/4, Lemma 4 and Theorem 3 fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing gap found. The proof of Theorem 3 has the correct logical structure: if every continuous f had a convergent y_n, then for f=W the increments y_{4s}-y_{2s} would converge to 0 almost surely and hence in distribution, contradicting Lemma 4. Lemma 4 in turn depends only on the covariance identity E W(u)W(v)=u∧v and four explicit double sums; I re-derived all four (and E y_n^2 = 1/2) and the limit E(y_{4s}-y_{2s})^2 -> 1/4 checks out. The only flaw I found is typographical: in the displayed computation of E x_{4s+1}x_{2s+1}, the second inner sum is printed with upper limit 4s-1, whereas the stated final value corresponds to the correct upper limit 4s; this does not affect the mathematics.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the sequence y_n = x_{n+1} - x_n, where x_n is the sum of f(k/n) for k=1,...,n-1 and f is continuous on [0,1]. Proposition 1 shows that if f is absolutely continuous, then y_n converges to the integral of f. Proposition 2 shows that whenever y_n converges, its limit must be that integral. The main result, Theorem 3, answers Furdui's Problem 1.32(b) by showing that there exist continuous functions for which y_n does not converge. The proof is probabilistic: taking f to be a Brownian path W, Lemma 4 computes the limiting variance of y_{4s} - y_{2s} as 1/4, which contradicts convergence of y_n, since convergence would force this difference to vanish in distribution. The paper also proves Propositions 1 and 2, the latter by the Stolz-Cesaro theorem.","tokens_in":3927,"tokens_out":1686,"duration_ms":17399,"significance":"If correct, the paper settles an open problem in the negative, showing that continuity alone does not guarantee convergence of the Riemann-sum increments y_n. The result is interesting and concise, and the probabilistic method is well chosen: it converts a deterministic existence question into a variance computation for a Gaussian process. The proof is essentially self-contained, relying only on standard Brownian motion facts, covariance calculations, L1 density of continuous functions, and the Stolz-Cesaro theorem. The explicit covariance computations in Lemma 4 are a strength, and the distinction between Propositions 1 and 2 is clean and illuminating.","major_comments":[],"minor_comments":[{"comment":"In the displayed computation of E x_{4s+1}x_{2s+1}, the upper limit of the second inner sum is printed as 4s-1, but the stated final value (32s^3+14s^2)/(12s+3) corresponds to the upper limit 4s; this is a typographical error only and does not affect the mathematics.","section":"Proof of Lemma 4"},{"comment":"The phrase 'direct calculations' covers four nontrivial double-sum evaluations; since these identities are the computational heart of the lemma, expanding at least one of them, or stating that they can be verified by a computer algebra system, would improve verifiability and readability.","section":"Proof of Lemma 4"},{"comment":"The statement 'without loss of generality we may assume that all realizations of W are everywhere continuous' is slightly imprecise: standard Brownian motion has a continuous modification, and one should formally work with that modification. The intended meaning is clear, but a one-sentence clarification would remove any ambiguity.","section":"Proof of Theorem 3"},{"comment":"Equation (10) identifies I_n(g) with f(n/(n+1)) - f(0); the reader must recall the fundamental theorem for absolutely continuous functions. Adding a brief parenthetical would make the step more transparent, though it is not logically problematic.","section":"Proposition 1"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis short note answers Furdui's Problem 1.32(b): for merely continuous f, the Riemann-sum increments y_n = x_{n+1} - x_n need not converge. The proof is probabilistic—take f to be a Wiener path—and it is the first counterexample I know, so the result is genuinely new.\n\nThe paper does several things well. Proposition 1 shows the limit equals the integral for absolutely continuous f, and Proposition 2 shows any convergent y_n must converge to the integral, so the target is exactly right. Theorem 3 follows from Lemma 4 via the standard almost-sure/in-distribution contradiction. The moment computations are explicit, and the variance limit 1/4 checks out. I re-derived the four covariance sums and the algebra is correct. The nonconstructive nature is stated plainly, and the author notes that an explicit construction remains open; that is a limitation, not a flaw.\n\nSoft spots are minor. The 'direct calculations' in Lemma 4 are compressed but complete enough to verify. One typo appears in the displayed formula for E x_{4s+1}x_{2s+1}: the second inner sum is printed with upper limit 4s-1, whereas the stated result corresponds to 4s. This is purely typographical. The proof's reliance on the existence of a standard Wiener process is a standard external theorem, not a circular step. The cited literature is appropriate.\n\nThe result is narrow—it settles one open problem in classical analysis without opening a new technique or reorganizing the area. But it is a clean, self-contained negative answer, and the probabilistic method here is a nice surprise. I think it deserves a serious referee and, likely, acceptance. I'd take it for a reading group for the trick, though I probably wouldn't cite it in my own immediate work.","headline":"Short, self-contained negative answer to an open problem about Riemann sums; the probabilistic argument is sound, with only a typographical blemish in the moment calculation.","tokens_in":4423,"tokens_out":1871,"would_cite":false,"duration_ms":16620,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["26A06","26A27","26A42","26A46","60Exx","60G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For merely continuous functions, the differences of successive Riemann sums need not converge, answering an open problem in the negative; the proof uses Brownian motion.","keywords":["Riemann sums","continuous functions","Brownian motion","Wiener process","probabilistic method","absolute continuity","limits of sequences","non-constructive existence proof"],"falsifier":"Recompute the four cross-moment sums in Lemma 4 and check whether $E(y_{4s}-y_{2s})^2$ tends to $1/4$; if the limit is actually $0$, the subsequence differences could vanish and the contradiction would disappear. A concrete numerical check is to simulate Brownian paths on the grid $\\{0,1/(4s),\\ldots,1\\}$, compute $y_{4s}-y_{2s}$ for large $s$, and estimate its variance: values approaching $1/4$ confirm the central calculation, while values approaching $0$ would refute it.","tokens_in":3592,"feed_emoji":"🎲","tokens_out":16220,"duration_ms":146676,"temperature":0.7,"pith_summary":"The paper settles an open question about the difference between successive Riemann sums. For a continuous $f$ on $[0,1]$, the sums $x_n=\\sum_{k=1}^{n-1} f(k/n)$ approximate $\\int_0^1 f$, and the question is whether the differences $y_n=x_{n+1}-x_n$ must converge. The paper proves two boundary results: for absolutely continuous $f$, $y_n\\to\\int_0^1 f$, and if $y_n$ converges for any continuous $f$, the only possible limit is that integral. Then it proves the main theorem: there exist continuous $f$ for which $\\lim_n y_n$ does not exist. The proof is non-constructive and uses Brownian motion as a continuous random function, showing that its subsequence differences $y_{4s}-y_{2s}$ converge in distribution to a centered normal law with variance $1/4$.","feed_headline":"Riemann sums can fail to converge for continuous f","feed_subtitle":"A Brownian-motion proof shows the first differences of discrete Riemann sums can oscillate without a limit.","key_machinery":"The load-bearing object is the standard Wiener process, or Brownian motion, $W=(W(u))_{u\\in[0,1]}$: a random continuous function whose covariance identity $E W(u)W(v)=u\\wedge v$ turns lattice sums of its values into exact polynomial expressions in $n$. The proof uses this identity to compute $E x_n^2$, $E x_nx_{n+1}$, and the four cross-moments displayed in Lemma 4, obtaining $E(y_{4s}-y_{2s})^2\\to 1/4$. Since a sequence of differences that converged almost surely to $0$ would converge in distribution to $0$, this non-degenerate normal limit is what forces the existence of a continuous counterexample.","core_discovery":"The central discovery is that continuity is not a sufficient condition for convergence of the differences of consecutive Riemann sums. The paper answers the open problem in the negative by proving existence of continuous $f:[0,1]\\to\\mathbb{R}$ with $\\lim_{n\\to\\infty} y_n$ nonexistent, while also proving $y_n\\to\\int_0^1 f(x)\\,dx$ whenever $f$ is absolutely continuous and that any existing limit must equal that integral. The existence proof is probabilistic: choosing $f$ to be a Brownian path gives an almost surely continuous function for which $y_{4s}-y_{2s}$ converges in distribution to $N(0,1/4)$, which cannot happen if $y_n\\to\\int f$ almost surely. Thus continuity of a function is genuinely not enough; some stronger smoothness, such as absolute continuity, is needed.","pith_inferences":["Editorial inference: the proof actually indicates a stronger statement: under the measure on continuous functions induced by Brownian motion, almost every function is a counterexample, with convergence of $y_n$ occurring only on a set of paths of probability zero.","Editorial inference: the same covariance computations could be repeated with other subsequence ratios, such as $y_{rs}-y_s$, likely producing normal limits whose variance depends on $r$; that would show the oscillation is not confined to one pair of subsequences.","Editorial inference: since Brownian paths are nowhere differentiable and fail Hölder continuity of every order at least $1/2$, the smoothness threshold for convergence may lie at some fractional regularity; a testable conjecture is that Hölder continuity of order greater than $1/2$ guarantees convergence.","Editorial inference: a deterministic counterexample might be constructed from a Weierstrass-type trigonometric series with carefully chosen frequencies, since the probabilistic argument proves existence but gives no explicit function."],"forward_implications":["The open problem is answered in the negative: continuity alone does not force $y_n$ to have a limit.","For absolutely continuous $f$, the limit always exists and equals $\\int_0^1 f(x)\\,dx$, and for any continuous $f$ whose sequence $y_n$ converges, that integral is the only possible limit.","The proof is non-constructive, so explicitly exhibiting a continuous function with $\\lim_n y_n$ nonexistent remains open.","Any sufficient condition for convergence of $y_n$ must be strictly stronger than continuity; absolute continuity is one such condition."],"supporting_citations":[{"why":"Identifies the open problem and defines the sequences $x_n$ and $y_n$; this is the question the paper answers.","marker":"[3]"},{"why":"Supplies the existence and continuity of Brownian motion sample paths used to manufacture the random continuous function $f=W$.","marker":"[4]"},{"why":"Supplies the density of continuous functions in $L^1$ used in the proof of Proposition 1.","marker":"[1]"},{"why":"Provides the difference-quotient convergence theorem used to prove Proposition 2.","marker":"[5]"},{"why":"Provides the original form of that difference-quotient theorem also used in Proposition 2.","marker":"[2]"}],"fun_headline_variants":["Continuous f can make Riemann sums oscillate","Brownian motion proves Riemann sums need smoothness","Riemann sum limit fails for continuous functions","Open problem solved: continuity is not enough","Riemann sums: continuity insufficient, absolute continuity works"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the standard theorem that a Brownian motion exists with continuous sample paths and covariance $E W(u)W(v)=u\\wedge v$; if that existence theorem were not available, the construction of the random continuous counterexample would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Continuous f can make Riemann sums oscillate","Brownian motion proves Riemann sums need smoothness","Riemann sum limit fails for continuous functions","Open problem solved: continuity is not enough","Riemann sums: continuity insufficient, absolute continuity works"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000238,"raw_usage":{"total_tokens":1387,"prompt_tokens":697,"completion_tokens":690,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":313,"completion_tokens_details":{"reasoning_tokens":621}},"tokens_in":313,"tokens_out":690,"duration_ms":6317,"temperature":1.0,"reasoning_tokens":621,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-27T21:54:28.735830+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the four cross-moment sums in Lemma 4 and check whether $E(y_{4s}-y_{2s})^2$ tends to $1/4$; if the limit is actually $0$, the subsequence differences could vanish and the contradiction would disappear. A concrete numerical check is to simulate Brownian paths on the grid $\\{0,1/(4s),\\ldots,1\\}$, compute $y_{4s}-y_{2s}$ for large $s$, and estimate its variance: values approaching $1/4$ confirm the central calculation, while values approaching $0$ would refute it.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the open problem and defines the sequences $x_n$ and $y_n$; this is the question the paper answers."},{"cited_title":"M¨ orters and Y","cited_arxiv_id":null,"evidence_quote":"Supplies the existence and continuity of Brownian motion sample paths used to manufacture the random continuous function $f=W$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the difference-quotient convergence theorem used to prove Proposition 2."},{"cited_title":"Ces` aro","cited_arxiv_id":null,"evidence_quote":"Provides the original form of that difference-quotient theorem also used in Proposition 2."}],"review_version":1}