Assume for all x belongs to real such that f(x0)=0.
Q2 show f(x)=2x is injective but not surjective.
Q3 show g(x)=x/2 if x is even g(x)=(x-1)/2 if x is odd is surjective but not injective.
Q4 Suppose f:R-->R is a function satisfying f(a+x)=f(a-x) and f(b+x)=f(b-x) for all x, where a,b are constants and a>b. Let w=2(a-b). Show that w is a period of f, i.e. f(x+w)=f(x) for all x belongs to R.
Q5 Suppose g:R-->R is a periodic function with period T>0 satisfying g(x)=f(-x) for all x. Show that there exists c with 0<c<T such that g(c+x)=f(c-x) for all x.
更新1:
sorry i missed part of the question. Q1 Let f: R --->R be a real function such that f(x+y)=f(x)f(y) for all x, y belongs to R. if f(x) is not identically equal to zero, show f(0)>0. Hint: Assume for all x belongs to real such that f(x0)=0.
更新2:
Q2 why not surjective?
更新3:
Q5 why g(x-c) =f(-(x-c)) =f(c-x) ??