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Continuous and differentiable function 
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#1
Feb1312, 06:09 PM

P: 19

function f:R>R can be written as a sum f=f1+f2 where f1 is even and f2 is odd。then 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
i am quiet confusing this statement , if f1 is continuous f2 is not how their sum to be continuous and differentiable as well. but i am sure this statement is true. can someone explain to me ??? 


#2
Feb1312, 07:46 PM

P: 446

I don't know if you could have [itex]f_1[/itex] continuous everywhere and [itex]f_2[/itex] discontinuous. A simple way that you could have two discontinuous functions sum to a continuous one is if you take cosine from [itex]\infty<\theta\leq 0[/itex] and 0 when greater than zero as [itex]f_1[/itex] and then a similar thing but where it is defined on the positive number and 0 for the negatives (and at 0 or else it will still be discontinuous). Individually there is a discontinuity at 0, but when you add them they are continuous.
This is a trivial case, but I hope it makes it clear. 


#3
Feb1412, 03:06 PM

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