Fourier Series

f 3 x 1 0 ≤ x 1 3 1 x ≤ 2Using the definitions of continuity and piecewise continuity given above: all three arecontinuous throughout 0,2, except at x 1 where each is piecewise continuous.Furthermore, the discontinuity at x 1 means that the integral of each function over 0,2is obtained by summing the contributions from 0,1 and 1,2, i.e.valid for m 1,2,3.02fm x dx 01fm x dx 12fm x dx 011 dx 123 dx 4The difference between the functions occurs at x 1 and it is seen that f 1 1 1,f 2 1 3but f 3 1 is not defined. This identifies an important point about discontinuous functions:they may or may not be defined at the point of discontinuity.A restriction is now imposed to ensure that only intervals of a certain type are allowable. Itis assumed that the functions are defined upon a closed interval of the real axis −a,a, anopen interval of the real axis −a,a or the infinite interval −,; the key requirement isthat the interval possesses x 0 as its centre point. The definitions are presented in termsof the closed interval but clearly hold for symmetric intervals that are open or infinite.Even functionAn even function is defined for all x ∈ −a,a to possess the property thatfx f−x .Thus the function is symmetric about the y-axis; one example is fx x 2 and another isfx cosx.Odd functionAn odd function is defined for all x ∈ −a,a to possess the property thatfx − f−x .Thus the function is anti-symmetric about the y-axis; one example is fx x 3 and anotheris fx sinx.These definitions do not impose conditions upon the behaviour of the function at thecentre-point x 0. If continuity is sought, this does not introduce any implications for aneven function beyond the basic property f0 − 0 f0 f0 0 but an odd function can2