1. ## Continuity

1.Using $\displaystyle \varepsilon-\delta$ definition,prove that f(x)=cosx is continuous.

2. Originally Posted by roshanhero
1.Using $\displaystyle \varepsilon-\delta$ definition,prove that f(x)=cosx is continuous.
What would using the $\displaystyle \epsilon - \delta$ definition achieve.

It's pretty obvious from definition of cosine as the horizontal distance made from the unit circle that $\displaystyle \cos{x}$ is continuous for all possible angles $\displaystyle x$...

3. I want to prove it from the formal definition as well.

4. Then if you want to prove it then do it.

I'll do it for you because I want to do it.

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It's easy to prove that $\displaystyle |\sin x-\sin y|\le|x-y|$ for each $\displaystyle x,y\in\mathbb R,$ in particular put $\displaystyle y=0$ and we get this useful inequality $\displaystyle |\sin x|\le|x|.$ As for the sake of the problem given $\displaystyle \epsilon>0,$ then for $\displaystyle 0<\delta=\epsilon,$ we have

$\displaystyle \left| \cos (x)-\cos (a) \right|=\left| 2\sin \left( \frac{x+a}{2} \right)\sin \left( \frac{x-a}{2} \right) \right|\le 2\left| \sin \left( \frac{x-a}{2} \right) \right|,$ hence $\displaystyle |\cos (x)-\cos(a)|=|x-a|<\epsilon.$

As required.