Does $L_1$ convergence of continuous functions imply pointwise convergence?

Question

Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $x\in[0,1]$?

I'm pretty sure the answer is no, but I'm not aware of a counterexample.

Answer 1

Consider $f_n(x) = (-1)^n x^n.$ Then $f_n\in C[0,1],$ and $f_n \to 0$ in $L^1,$ but $f_n(1)$ does not converge.

Answer 2

The "dancing indicators" counterexample: $$\chi_{[0, 1/2]}, \chi_{[1/2, 1]}, \chi_{[0, 1/3]}, \chi_{[1/3, 2/3]}, \chi_{[2/3, 1]}, \chi_{[0, 1/4]}, \chi_{[1/4, 2/4]}, \ldots$$

Edit: as noted in the comments, these functions are not continuous. However, if you replace the indicators with continuous "tent" functions supported on the same interval, you obtain the same contradiction. I like this type of counterexample more than the one given by zhw since you can use it to construct a sequence that fails to converges pointwise for all $x \in [0,1]$, rather than one that fails to converge pointwise at a single $x$.