= Shrinking continuous spikes with vanishing integral
{title2=$f_n(x)=\max(1-n^2|x-1/n|,0)$}
For $n\ge2$, these <continuous functions> on $[0,1]$ have maximum one and integral $1/n^2$. Their supports shrink towards zero while their value at zero remains zero. They converge pointwise to zero and their integrals converge to zero, but their convergence is not <uniform convergence>. Thus convergence of integrals and continuity of the pointwise limit do not imply <uniform convergence>.
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