Also sprach Zarathustra Dec 2009 1,506 434 Russia May 21, 2010 #2 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.
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.