Odd extension of function

Question

Say there is a function defined in [0,a), and its value at x=0 is not zero then how can I find odd extension, as from what I know, "odd extension makes function odd and odd function takes value 0 at x=0".

Answer 1

An odd function has to have $f(0)=0.$ The rule for an odd function is that $f(-x) = -f(x)$ for every $x$. Suppose $f(0) = b$. If you plug $0$ into the rule, you get

$$ f(-0) = -f(0)$$

or

$$f(0) = -f(0)$$

or

$$b= - b$$

So you have to have $b=0$.

However, in some contexts, e.g., Fourier series, that one value at $x=0$ doesn't matter. You could take a function like $f(x) = 1-x$ on the interval $[0,1)$ and form the odd extension, ignoring the fact that $f(0) = 1$. Then find the Fourier sine series. Since $f(x)$ is discontinuous at $x=0$, the series won't converge to the right value. It will converge to the average of the left and right hand limits, which has to be $0$ for an odd function. In short, the Fourier series will be $0$ at $0$, even if $f(0)$ is not.