Suppose I have the piecewise function $f(x) = \begin{cases}1, & 0<x<1 \\ 0 & x>1 \end{cases}$
and I wanted to extend it to an even function over the entire real line. How would I do that?
I know that normally for a function $\phi$ defined on the half line, the even extension would normally be $\phi_{\text{even}}(x)=\begin{cases} \phi(x)& x\geq 0 \\ +\phi(-x)& x\leq 0\end{cases}$, but how do I take care of a case when the function is constant on each piece, and thus has no "$x$'s" to substitute in a negative for?
I don't have a prof to ask - I'm trying to learn this on my own, so please don't be snarky if I sound clueless!
Thank you ahead of time.
The extension is not unique since it is not known that what are $f(0)$ and $f(1)$. On the other hand, all you need is defined for $x<0$, $$ f(x):=f(-x). $$