Decay of a nonnegative uniformly continuous integrable function ($f\ge 0,{\int }_0^{\infty }f<\infty ⟹f(t)\to 0$)

A nonnegative uniformly continuous function on $[0,\infty )$ with finite integral tends to zero. Otherwise uniform continuity gives disjoint intervals of a fixed width on which the function exceeds a fixed positive value, contradicting integrability. This supplies a useful dissipation argument for Lyapunov functions with merely continuous vector fields.