# Independency Preseving of two Independent random variables

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 $$a(X)=\max\{X,0\}$$.

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