Let f be a bounded function on [a, b], which is continuous everywhere except finitely many...

 

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Let f be a bounded function on [a, b], which is continuous everywhere except finitely many points X1, X2, follows. We'll use induction on N. XN. We'll prove that f is integrable as (a) Base case: N = 1. So f has only one discontinuity in [a, b] - call it x₁. We'll simplify our proof by assuming that x₁ = (a, b). Let > 0. We'll show that there exists a partition Q of [a, b] so that UQ(f) - Lo(f) < € (i) Let 0 < a < min{a-xil. lb-x11). Choose a partition P = P₁ U P₂ where P₁ is a partition of [a, x₁-a] and P2 is a partition of [x+a, b]. Sketch such a partition. (ii) Let M = sup{f(x) | xe [x₁ - a x₁+a]) and m = inf{f(x) | xe [x-a, x+a]). Show that you can choose a so that MAX - MAX SE/3, where Ax is the length of the interval [x₁-a, x₁ + a]. (iii) Use the fact that f is continuous on [a, x₁-a] and on [x₁+a, b] to show that we can choose P1, P2 so that Up (f) - Lp1 (f)</3 and Upz (f) - Lez (f) <e/3. (iv) Use (ii) and (iii) to conclude that there exists a partition Q of [a, b] so that Up (f)- Lo(f)<<, and deduce that f is integrable on [a, b]. (b) Now assume the result holds if f has N - 1 discontinuities, and let f have N discontinuities x1,..., XN (Again we'll simplify things so that none of the discontinuities are at the endpoints.) Let 0 < a <min{XN-XN-11. Ib-xN 1). Choose a partition P = P₁ U P₂ where P₁ is a partition of [a, XN - a] and P₂ is a partition of [XN + a, b]. (i) Given € > 0, show that we can choose P₁ so that Up - Lp1 </3 and P2 so that Up2 - Lp2 < €/3. (ii) Similar to (a - ii): Let M = sup{f(x) | xe [XN -a, XN +a]) and m = inf{f(x) | xe [XN-a, xx+a]). Show that you can choose a so that MAX - MAX SE/3. (iii) Conclude that f is integrable on [a, b]. Let f be a bounded function on [a, b], which is continuous everywhere except finitely many points X1, X2, follows. We'll use induction on N. XN. We'll prove that f is integrable as (a) Base case: N = 1. So f has only one discontinuity in [a, b] - call it x₁. We'll simplify our proof by assuming that x₁ = (a, b). Let > 0. We'll show that there exists a partition Q of [a, b] so that UQ(f) - Lo(f) < € (i) Let 0 < a < min{a-xil. lb-x11). Choose a partition P = P₁ U P₂ where P₁ is a partition of [a, x₁-a] and P2 is a partition of [x+a, b]. Sketch such a partition. (ii) Let M = sup{f(x) | xe [x₁ - a x₁+a]) and m = inf{f(x) | xe [x-a, x+a]). Show that you can choose a so that MAX - MAX SE/3, where Ax is the length of the interval [x₁-a, x₁ + a]. (iii) Use the fact that f is continuous on [a, x₁-a] and on [x₁+a, b] to show that we can choose P1, P2 so that Up (f) - Lp1 (f)</3 and Upz (f) - Lez (f) <e/3. (iv) Use (ii) and (iii) to conclude that there exists a partition Q of [a, b] so that Up (f)- Lo(f)<<, and deduce that f is integrable on [a, b]. (b) Now assume the result holds if f has N - 1 discontinuities, and let f have N discontinuities x1,..., XN (Again we'll simplify things so that none of the discontinuities are at the endpoints.) Let 0 < a <min{XN-XN-11. Ib-xN 1). Choose a partition P = P₁ U P₂ where P₁ is a partition of [a, XN - a] and P₂ is a partition of [XN + a, b]. (i) Given € > 0, show that we can choose P₁ so that Up - Lp1 </3 and P2 so that Up2 - Lp2 < €/3. (ii) Similar to (a - ii): Let M = sup{f(x) | xe [XN -a, XN +a]) and m = inf{f(x) | xe [XN-a, xx+a]). Show that you can choose a so that MAX - MAX SE/3. (iii) Conclude that f is integrable on [a, b].

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a Base case N 1 1 Choose a partition P PUP where P is a partition of a a1 a and P is ... View the full answer