Uniform Continuity of Integral Functions in (0,1)

[daniel_i_l]

Homework Statement

Prove that the function $$\int^{1}_{x}\frac{sin t}{t}dt$$ 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 $$G(x) = \int^{x}_{1}f(t) dt$$ Is defined and continues in [0,1] so it's uniformly continues in (0,1). But in (0,1)
$$G(x) = \int^{x}_{1}\frac{sin t}{t}dt$$ And since that's UC in (0,1), then also $$-\int^{x}_{1}\frac{sin t}{t}dt = \int^{1}_{x}\frac{sin t}{t}dt$$ 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.

 

[Dick]

I think if you want f(x)=sin(x)/x to be continuous, you'll want to define f(0)=1. Other than that, there is nothing wrong with proving it that way. If the integral is continuous on [0,1], then it is uniformly continuous on (0,1). I don't know why you are messing around with changing the limits of integration though.

 

[daniel_i_l]

Yeah, it's supposed to be f(0)=1. Thanks for your confirmation.