(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that the function [tex]\int^{1}_{x}\frac{sin t}{t}dt[/tex] is uniformly continues in (0,1).

2. Relevant equations

3. 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Integrals and continuity

**Physics Forums | Science Articles, Homework Help, Discussion**