Is the following statement true? If ∫₀^∞ f converges then f(x) → 0

No. f with a spike of height n and width 2/n³ at each integer n The total area is finite but f is unbounded, so it does not tend to 0.

Counterexample

Is this true?

If $\int_{0}^\infty f$ converges then f(x) → 0

No — it is false.

The counterexample

f with a spike of height n and width $2/n^{3}$ at each integer n

Why it fails

The total area is finite but f is unbounded, so it does not tend to 0.

Keep practicing

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.