Independency Preseving of two Independent random variables

Question

Suppose we have two independent random varaibles $X_1$ and $X_2$. And we have a function $a(\cdot)$.

Are the two new random variables $a(X_1)$ and $a(X_2)$ still independent? For example, $a(X)=X-3$ or $a(X)=\max\{X,0\}$.

Answer 1

If $X$ and $Y$ are independent and $f$ is any measurable function from $\mathbb R$into itself then $f(X)$ and $f(Y)$ are independent. This follows easily from definition of independence.

In particular this holds when $f$ is continuous. In your examples the functions are continuous