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daniel_i_l
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Homework Statement
Prove that the function [tex]\int^{1}_{x}\frac{sin t}{t}dt[/tex] is uniformly continues in (0,1).
Homework Equations
The Attempt at a Solution
First if all, I defined f(x) as sin(x)/x for x=/=0 and 0 for x=0. So f is continues in [0,1]. Now [tex]G(x) = \int^{x}_{1}f(t) dt[/tex] Is defined and continues in [0,1] so it's uniformly continues in (0,1). But in (0,1)
[tex]G(x) = \int^{x}_{1}\frac{sin t}{t}dt[/tex] And since that's UC in (0,1), then also [tex]-\int^{x}_{1}\frac{sin t}{t}dt = \int^{1}_{x}\frac{sin t}{t}dt[/tex] is UC in (0,1). Is that enough?
Because in the book they used the fact that G(x) has a derivative in [0,1] and that G'(x) = f(x). Then they used the fact that |f(x)|<=1 (meaning that the derivative of G is bounded) to prove that G(x) is UC.
What's wrong with my way?
Thanks.