= Concentrating triangular bump
{title2=$f_n(x)=n\max\{1-|nx-1|,0\}$}
On $[0,1]$, for $n\geq2$, these <continuous functions> have support $[0,2/n]$, height $n$, and integral one. Their <pointwise limit> is zero: at zero every value vanishes, while every fixed positive argument eventually leaves the support. Thus <pointwise convergence> of <continuous functions> on a <compact interval> does not suffice for convergence of their integrals. The mass becomes more concentrated as its height grows.
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