1. The problem statement, all variables and given/known data Prove functions f and g are continuous in ℝ. It's known that: i) lim g(x)=1, when x approaches 0 ii)g(x-y)=g(x)g(y)+f(x)f(y) iii)f2(x)+g2(x)=1 3. The attempt at a solution g(0) has to be equal to 1 because it's known that lim g(x)=1, when x approaches 0. Otherwise g won't be continuous at x=0. g(0)=1 implies that f(0)=0 (iii) And if g(0)=1 and f(0)=0 then g(-y)=g(y) (ii) so g is even function. So it looks like g(x)=sin (x) and f(x)=cos(x). However, g and f may not be unique. f+g is continuous if f and g are continuous but not vice versa The definition of continuity is ∀ε>0∃δ>0: |x-x0|<δ ⇒|f(x)-f(x0|<ε ,x0∈ℝ How to proceed?