Can a fourier series be discontinous?

Yes, a Fourier series can represent a function that is discontinuous. While the series converges to the function at points of continuity, at points of discontinuity, it converges to the average of the left-hand and right-hand limits. This phenomenon is known as the Gibbs phenomenon, where the series may exhibit oscillations near the discontinuities. Despite these oscillations, the Fourier series still provides a useful approximation of the function.