The total square error of a function g(x) relative to f (x) is de ned to be the integral. If f (x) is piecewise continuous on [ , ], we get. [ f (x) ]2 dx =< f, f > (called bessel"s inequality) and this shows that. Remark: (uniqueness theorem) let f (x) and g(x) be piece- wise continuous in the interval [ , ] and have all the same fourier coe cients. Then f (x) = g(x) except, perhaps, at points of discon- tinuity. De nition: a curve (or path) in rn is a function. : [a, b] r rn. We usually call the image of the curve and the function a parametrization of the curve. Is continuous at c (a, b) if lim t c. (t) = (c). if and only if the components i(t), i = 1, 2, , n, are continuous at c.