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This was disproved by Paul du Bois-Reymond, who showed in 1876 that there is a continuous function whose Fourier series diverges at one point.
Edvard Henrik Carleson (16 November 1820 – 1 April 1884) was a Councilor of Justice and a Councilor of State of the Kingdom of Sweden, etc., son of Jacob Edvard Carleson, Lieutenant Colonel of the Swedish Army, and wife Liboria Fredrika Eleonora Harmens.
The case p=2 is due to Lennart Carleson, and for this reason the general result is called the Carleson-Hunt theorem.