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Riemann’s example of function f for which exists for all x, but is not differentiable when x is a rational number with even denominator. Riemann’s example of function f for which exists for all x, but is not differentiable when x is a rational number with even denominator. What does a derivative look like? Can we find a function that can’t be a derivative but which can be integrated? Does a derivative have to be continuous? If F is differentiable at x = a, can F '(x) be discontinuous at x = a? If F is differentiable at x = a, can F '(x) be discontinuous at x = a? Yes! x 2 sin 1 , x 0, x Fx x 0. 0, x 2 sin 1 , x 0, x Fx x 0. 0, F' x 2xsin 1 x cos 1 x , x 0. x 2 sin 1 , x 0, x Fx x 0. 0, F' x 2xsin 1 x cos 1 x , x 0. F h F 0 F' 0 lim h 0 h h 2 sin 1 h lim h0 h lim h sin 1 h 0 . h0 x 2 sin 1 , x 0, x Fx x 0. 0, F' x 2xsin 1 x cos 1 x , x 0. F h F 0 F' 0 lim h 0 h lim F' x does not exist, but 2 1 x0 h sin h lim F' 0 does exist (and equals 0). h0 h lim h sin 1 h 0 . h0 How discontinuous can a derivative be? Can it have jump discontinuities where the limits from left and right exist, but are not equal? How discontinuous can a derivative be? Can it have jump discontinuities where the limits from left and right exist, but are not equal? No! If lim f ' x and lim f' x exist, xc xc then they must be equal and they must equal f ' c. Mean Value Theorem: f x f c f ' c lim lim f ' k, x k c x c x c x c f x f c f ' c lim lim f ' k, c k x x c xc x c If lim f ' x and lim f' x exist, xc xc then they must be equal and they must equal f ' c. Mean Value Theorem: f x f c f ' c lim lim f ' k, x k c x c x c x c f x f c f ' c lim lim f ' k, c k x x c xc x c The derivative of a function cannot have any jump discontinuities! Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series b Defined a f x f xdx as limit of i xi xi1 Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series b Defined a f x f xdx as limit of i xi xi1 Key to convergence: on each interval, look at the variation of the function Vi sup x[ xi1, xi ] f x inf x[x i1 ,x i ] f x Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series b Defined a f x f xdx as limit of i xi xi1 Key to convergence: on each interval, look at the variation of the function Vi sup x[ xi1, xi ] f x inf x[x i1 ,x i ] f x Integral exists if and only if Vi xi xi1 can be made as small as we wish by taking sufficiently small intervals. Any continuous function is integrable: Can make Vi as small as we want by taking sufficiently small intervals: Vi xi xi 1 ba xi xi1 ba b a . Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series Riemann gave an example of a function that has a jump discontinuity in every subinterval of [0,1], but which can be integrated over the interval [0,1]. Riemann’s function: f x n1 n x n2 x nearest integer , when this is 1 2 , x 1 0, when distance to nearest integer is 2 –2 –1 n x n2 1 2 has n jumps of size 2 between 0 and 1 n 2 2 a At x 2b , gcda,2b 1, the function jumps by 8b 2 Riemann’s function: f x n1 n x n2 2 a At x 2b , gcda,2b 1, the function jumps by 8b 2 The key to the integrability is that given any positive number, no matter how small, there are only a finite number of places where is jump is larger than that number. Riemann’s function: f x n1 n x n2 2 a At x 2b , gcda,2b 1, the function jumps by 8b 2 The key to the integrability is that given any positive number, no matter how small, there are only a finite number of places where is jump is larger than that number. Conclusion: Fx 0 f t dt exists and is well - defined x for all x, but F is not differentiable at any rational number with an even denominator.

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