Property of the characteristic function

Question

Hello is the first time I encounter the characteristic function $\chi_A(x)=1$ for $x\in A$ and $\chi_A(x)=0$ for $x\notin A$. I have proved some of the properties of $\chi_A(x)$ but I'm struggling to proof that $$\chi_{[x,a]}(y)=\chi_{[0,y]}(x)$$ where $a$ is a constant and $0 \leq x \leq y \leq a$.

Answer 1

You have enough information now to compute each value directly!

What is $\chi_{[x, a]}(y)$? HINT: by assumption $x\le a\le y$, so what do you know about $y$ and the set $[x, a]$?

Similarly, what is $\chi_{[0, y]}(x)$? HINT: by assumption, $0\le x\le y$. . .