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Prove the following facts:?
A) |x| is integrable on [-1,2]
b) x^(1/4) is integrable on [0,9]
c) the function h(x)= x^2 x∈[0,1]
2x+3 x∈(1,2]
is integrable on [0,3]
d) if f, g are integrable on [a,b] the so is f-g
I would really appreciated if anyone can help me with this, thank you!
1 Answer
- ?Lv 78 years agoFavorite Answer
A function is Riemann integrable over a compact interval if, and only if, it's bounded and continuous almost everywhere on this interval (with respect to the Lebesgue measure). Since {x_1,...x_n} is finite, it's Lebesgue measure is 0, so that g is continuous almost everywhere on [a, b]. And since, in addition, g is bounded, it follows g is Riemann integrable on [a, b].
a) |x| is bounded and continuous on [-1, 2].
b) x^(1/4) is bounded and continuous on [0, 9]
c) Redefine h(x) = f(x) + g(x) where f(x) = x^2 on [0, 1] and g(x) = 2x+3 on (1, 2]. h(x) is bounded on [0, 3]. It is continuous on [0, 3] except at x = 1. Since it's only a finite number of points where the discontinuity exists, the Lebesgue Measure at x = 1 is 0. Consequently, it is integrable.
d) If f & g are integrable, then f - g is integrable because f is bounded and continuous (almost everywhere) and g is bounded and continuous (almost everywhere). So, f - g is bounded and continuous (almost everywhere).
