Can Any Function Be Decomposed into Odd and Even Subfunctions?

[thenewbosco]

Hello, the question here says:

Show that any given function can be decomposed into the sum of manifestly odd and even subfunctions.

What i have done is just assumed a continuous, differentiable function, with a number a in the domain of the function, then shown that a taylor series for a function alternates between even and odd functions as the powers of x change from even to odd numbers. Is this enough for this question or is there something i haven't seen?

thanks

 

[Hurkyl]

Is this enough for this question or is there something i haven't seen?

No! Surely you see that you have only proven it for functions that have Taylor series expansions at every point?

(Incidentally, even infinitely differentiabe functions can fail to have Taylor series)



This is the sort of problem where you just write down what things mean, and ideas should become evident. What does it mean for the function f(x) to be decomposed into an odd and an even subfunction? What does it mean for a function to be even? What does it mean for a function to be odd?

(Incidentally, you should always ask yourself questions like this anytime you get stuck. In fact, it usually helps to ask these questions before you get stuck)