Shrinking continuous spikes with vanishing integral ($f_n(x)=\max (1-n^2|x-1/n|,0)$)

For $n\ge 2$, 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.