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Sufficiency conditions for a `well-behaved' periodic function f(x) with period 2L to have a convergent Fourier series. These are that f must be bounded and have a finite number of maxima, minima and discontinuities for x ∈ [0,2L) ; then the Fourier series will converge to f wherever it is continuous, and at a jump discontinuity it will converge to the midpoint of the two one-sided limits. Not that these are not necessary conditions, for example f(x) = sin(1/x) does have a convergent Fourier series.