Continuous and differentiable function

[frankpupu]

Homework Statement

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

 

[Dick]

Homework Statement

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

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).

 

[HallsofIvy]

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

 

[frankpupu]

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

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?

 

[Dick]

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?

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?

 

[frankpupu]

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?

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

 

[Dick]

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

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).