convergence with respect to integral norm but not pointwise I want to give an example of a sequence of functions $f_1 \dots f_n$ that converges with respect to the metric $d(f,g) = \int_a^b |f(x) - g(x)| dx$ but does not converge pointwise.
I'm thinking of a function $f_n$ that is piecewise triangle, whose area converges to some constant function, but doesn't converge pointwise.
I just can't manage to formalize it.

You can suppose that $g(x) = 0$, because your metric is translation invariant ; $d(f_n,g) = d(f_n-g,0)$. Think of the sequence $f_n : [0,1] \to \mathbb R$ defined by $f_n(x) = x^n$ if $n$ is odd, and $(1-x)^n$ if $n$ is even. Therefore,
$$
d(f_{2n+1},0) = \int_0^1 (1-x)^{2n+1} \, dx = \frac 1{2n+2} \underset{ n \to \infty} {\longrightarrow} 0
$$
or
$$
d(f_{2n},0) = \int_0^1 x^{2n} \, dx = \frac 1{2n+1} \underset{ n \to \infty} {\longrightarrow} 0
$$
but $f_n$ does not converge pointwise at $0$ and $1$ because the values oscillates between $0$ and $1$.
I know my answer uses the idea of peaking "the triangles" alternatively at $0$ and $1$ of Brian, but I still think it's worth it to see a more-or-less "trivial" example (using not-defined-by-parts polynomial functions), so I kept my answer there anyway.
Hope that helps,
