Independent editorial note
NBHM Combined 2026 Question 40: why the printed assumptions permit a singleton
The official key marks assertion (b) false. Under the assumptions printed in the paper, a continuous counterexample shows that it is true. This note gives the construction and all convergence checks.
Mathematical conclusion
(a) True · (b) True as printed
Published-key response
(a) True · (b) False
What the item is testing
For a nonnegative continuous function on the nonnegative real line, consider the positive exponents for which its power has a finite integral. Hölder's inequality makes this exponent set convex: if two exponents belong to it, every exponent between them also belongs to it. Therefore every nonempty such set is an interval and is connected, proving assertion (a).
Connectedness alone does not exclude a one-point interval. Assertion (b) therefore needs a separate argument.
A continuous function with exponent set exactly {1}
Start with the slowly decaying baseline
b(x) = 1 / ((x + e)(log(x + e))²).
For every integer n ≥ 1, add a triangular continuous spike centred at 3n, of height eⁿ and half-width e⁻ⁿ/n². The supports are disjoint, so their sum is locally finite. Adding the baseline gives a nonnegative, continuous, nonzero function.
At t = 1
The baseline is integrable, and the spike areas sum as ∑1/n². The full integral is finite.
For 0 < t < 1
The baseline power has a tail comparable to x⁻ᵗ(log x)⁻²ᵗ, whose integral diverges.
For t > 1
The nth spike contributes 2eⁿ⁽ᵗ⁻¹⁾/((t+1)n²), so the spike series diverges.
The power integral is finite at t = 1 and infinite at every other positive exponent. The exponent set is therefore exactly {1}.
Why this is easy to miss
A convex subset of the real line is an interval, but an interval may be a singleton. A boundedness assumption would change the available implications between exponents; the printed problem includes no such assumption.
Source and review status
This is an independent InshiHub derivation, not an official correction. It was checked against the paper and published key on 27 July 2026. InshiHub is not affiliated with NBHM or IMSc.
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