No: the unbounded length of the domain matters. Let and define on . These are continuous functions with compact support, and , so the sequence has uniform convergence to zero. However, the change of variables gives
(1)
The integral of the limit is zero. The supports widen as the heights decrease; a uniform bound on the heights gives no bound on the total area over an infinite interval.
Figure 1.
Concentrating bumps, spreading bumps, and a one-sided approximate identity
. Left: the examples in part (b)(i) converge pointwise to zero while their integrals remain one. Middle: the examples in part (b)(iii) have uniform convergence to zero but retain integral one by spreading across the real line. Right: the nonnegative functions in part (c) concentrate their mass at zero and recover the endpoint value of a continuous function. Details.

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