= Solution
{c}
\b[No: the unbounded length of the <domain> matters.] Let $\phi(t)=\max\{1-|t|,0\}$ and define $f_n(x)=n^{-1}\phi(x/n)$ on $\mathbb R$. These are <continuous functions> with <compact support>, and $\sup_{\mathbb R}|f_n|=1/n\to0$, so the <sequence> has <uniform convergence> to zero. However, the <change of variables> $t=x/n$ gives
$$
\boxed{\int_{-\infty}^{\infty} f_n(x)\,dx=\int_{-1}^1(1-|t|)\,dt=1.}
$$
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>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/2023-convergence-and-integrals.png]
{title=Concentrating bumps, spreading bumps, and a <one-sided approximate identity>}
{description=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>.}
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