 Concentrating triangular bump

On $[0,1]$, for $n\ge 2$, 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.