Normal convergence and continuity of Fourier series

Question

For piecewise $C^1$, Lipschitz continuous function $f$ which is also $2\pi$ periodic we know that the Fourier series is normal convergent, but then this should imply also the continuity of the series since the series is a sum of continuous functions. And we know that the series converges on continuous points uniform to the function $f$ itself, but this sounds contradictory. It would be great if you guys can help me spot any errors in the reasoning above. Thanks

Answer 1

this should imply also the continuity of the series since the series is a sum of continuous functions

The sum of two continuous functions is continuous. The sum of a finite collection of continuous functions is continuous. But the sum of an infinite sequence of continuous functions does not need to be continuous, even if it converges pointwise everywhere.

For one example, consider the sequence of functions $f_n: \mathbb{R} \to \mathbb{R}$,

$$ f_0(x) = \begin{cases} 0 & x \leq 0 \\ x & 0 < x \leq 1 \\ 1 & x > 1 \end{cases} $$ $$ f_n(x) = \begin{cases} 0 & x \leq 0 \\ 2^{n-1} x & 0 < x \leq 2^{-n} \\ 1 - 2^{n-1} x & 2^{-n}< x \leq 2^{1-n} \\ 0 & x > 2^{1-n} \end{cases} \quad (n \geq 1 )$$

$$ \sum_{n=0}^N f_n(x) = \begin{cases} 0 & x \leq 0 \\ 2^N x & 0 < x \leq 2^{-N} \\ 1 & x > 2^{-N} \end{cases} $$

$$ \sum_{n=0}^\infty f_n(x) = \lim_{N \to \infty} \sum_{n=0}^N f_n(x) = \begin{cases} 0 & x \leq 0 \\ 1 & x > 0 \end{cases} $$

Every function $f_n$ is piecewise linear and continuous everywhere. The sum of the first $N$ functions is piecewise linear and continuous everywhere. But the infinite series is not continuous at $0$.