Verify that X^2 is a continuous random variable..

Question

Let X be a continuous random variable. Verify that the following function is continuous random variable. $X^2$.

From the definition of a continuous random variable, I know that $P(X=x)=0$ for all $x \in R$. I can't see how I'm gonna come up with a 0 and write it formally but I tried $P(X^2=x^2)=0$ if X is a continuous RV

Answer 1

$P(X^2=x) = P(X=-x^\frac12)+P(X=x^\frac12)$ for $x > 0$. $P(X^2=0)=P(X=0)$ .$P(X^2=x)=0$ for $x<0$.

Answer 2

From what I understood I can write my answer the following way: Since $P(X^2=x)$ with $x \in R$. x can be =0 , <0 and >0. In the first case, $P(X^2=0)=0$ which is $P(X=x)=0$. In the second case, $P(X^2=y)$ with y<0 which gives P(impossible)=0. And for the last one, $P(X^2=y)$ with y>0, $P( X=-squareroot(y) )+ P( X=sauarerroot(y) )=0 $ Since it's 0 in all these cases then from the definition X is a continuous random variable.