Is it possible to represent all odd and even functions by a sine and cosine series respecitively?

Question

My textbook [1] claims that

it is possible to represent all odd functions by a sine series and all even functions by a cosine series.

Is anyone able to provide a proof or justification for this?

[1] - Riley, K. F.; Hobson, M. P.; Bence, S. J., Mathematical methods for physics and engineering. A comprehensive guide (3rd ed.) Cambridge University Press. p. 416 (2006).

Answer 1

The function $$d(x) = \begin{cases} 1 & \text{$\frac{x}{2π}$ is rational} \\ 0 & \text{otherwise} \end{cases}$$

is even but can't be represented by a cosine series.

But I think this comment is probably correct:

Probably the book restricts the class of the functions. For example they are assumed to be continuous or at least piecewise-continuous

The book is “Mathematical methods for Physics and Engineering”. Functions that arise in physics and engineering are always continuous or nearly continuous. Monstrosities like $d$ above don't have any relevance to the physical universe.