Prove that f(x)=cos(x) is continuous for all x in R.

Ted

Feb 2010
240
64
China
Prove that f(x)=cos(x) is continuous for all x in R.
 
Dec 2009
1,506
434
Russia
Let we choose x0 and we choose e>0. WE need to prove that for all e>0 exist d>0 so if |x-x0|<d then |cosx-cox0|<e.

|cosx-cosx0|=|-2sin(x+x0/2)*sin(x-x0/2)|<2|(x-x0)/2|*1

So if |x-x0|<e then |cosx-cox0|<e, and it is enough to choose e=d.

We proved that f(x)=cosx is continuous at some point x0 in R ==> the function is continuous for all x in R.