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Continuous and differentiable function

by frankpupu
Tags: continuous, differentiable, function
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frankpupu
#1
Feb13-12, 06:05 PM
P: 19
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
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Dick
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Feb13-12, 06:10 PM
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Quote Quote by frankpupu View Post
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
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
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Feb13-12, 06:10 PM
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Given function f, let [itex]f_e(x)= (f(x)+ f(-x))/2[/itex] and [itex]f_o(x)= (f(x)- f(-x))/2[/itex].

frankpupu
#4
Feb13-12, 06:15 PM
P: 19
Continuous and differentiable function

Quote Quote by HallsofIvy View Post
Given function f, let [itex]f_e(x)= (f(x)+ f(-x))/2[/itex] and [itex]f_o(x)= (f(x)- f(-x))/2[/itex].
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
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Feb13-12, 06:20 PM
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Quote Quote by frankpupu View Post
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
#6
Feb13-12, 06:36 PM
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Quote Quote by Dick View Post
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
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Feb13-12, 07:32 PM
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Quote Quote by frankpupu View Post
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).


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