Continuous and differentiable function

1. Feb 13, 2012

frankpupu

1. The problem statement, all variables and given/known data
function f:R->R can be written as a sum f=f1+f2 where f1 is even and f2 is odd。show that if f is continuous then f1 and f2 may be chosen continuous, and if f is differentiable then f1 and f2 can be chosen differentiable

2. The attempt at a solution
i have try some examples, but i still cannot get the idea from that

2. Feb 13, 2012

Dick

You should be able to find an expression for f1 and f2 given f. Think about f(x)+f(-x) and f(x)-f(-x).

3. Feb 13, 2012

HallsofIvy

Staff Emeritus
Given function f, let $f_e(x)= (f(x)+ f(-x))/2$ and $f_o(x)= (f(x)- f(-x))/2$.

4. Feb 13, 2012

frankpupu

yes f(x)=((f(x)+f(−x))/2 )+(f(x)−f(−x))/2) then (f(x)+f(−x))/2 is even and (f(x)−f(−x))/2 is odd,then i don't know how to argue their continuity and differentiability?

5. Feb 13, 2012

Dick

I don't think you have to prove it from scratch. If f(x) is continuous then f(-x) is continuous. What theorem about continuous functions might you use to prove that?

6. Feb 13, 2012

frankpupu

if f(x)is continuous then f(-x)is continuous ,(f(x)+f(-x))/2 is continuous by sum rule (f(x)-f(-x))/2 is continuous as well .but they should be chosen continuous

7. Feb 13, 2012

Dick

You are given f(x) is continuous. There is no need to show that. You do need to show f(-x) is also continuous. That's a composition rule. f(-x)=f(g(x)) where g(x)=(-x).