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

Using the inequality [tex]|\sin(x)| < |x|[/tex] for [tex] 0 < |x| < \frac{\pi}{2}[/tex], prove that the sine function is continuous at 0.

2. Relevant equations

Definition of continuity: A function f: R -> R is continuous at a point [tex]x0 \in R[/tex], if for any [tex]\epsilon > 0[/tex], there esists a [tex]\delta(\epsilon; x0)[/tex] such that if [tex]|x - x0| \leq \delta(\epsilon; x0)[/tex], then [tex]|f(x) - f(x0)| \leq \epsilon[/tex].

3. The attempt at a solution

To be honest, I am not really sure about this. I know that in order to prove continuity at a point, I can take the limit of the function at that point and see if it's equal to the value of the function. With this method, though, I am not using the inequality given, and therefore I cannot think of a way to do this. In class we were given no examples, so I am really lost here. I tried using the definition of continuity, but I did not manage to get anywhere that made sense.

I am sure this cannot be hard and I am probably missing something obvious. If anyone could help I'd truly appreciate it. Thank you.

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# Homework Help: Continuity of sine function

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