The problem involves a function f: R -> R that can be expressed as the sum of an even function f1 and an odd function f2. The task is to demonstrate that if f is continuous, then f1 and f2 can also be chosen to be continuous, and similarly for differentiability.
Some participants have proposed expressions for f1 and f2 and are considering the implications of continuity and differentiability. There is ongoing exploration of relevant theorems related to continuous functions, but no consensus has been reached on the argumentation needed to support the claims about f1 and f2.
Participants note that if f is continuous, then f(-x) is also continuous, and they reference the sum and composition rules for continuous functions. However, there is a lack of clarity on how to formally establish the continuity and differentiability of the derived functions f1 and f2.
frankpupu said: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
HallsofIvy said: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 said: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 said: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 said: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