# Proving of Y=g(X) as a continuous random variable

#### bl00d

##### New member
If X is a continuous random variable and g is a continuous function
defined on X (Ω), then Y = g(X ) is a continuous random variable.
Prove or disprove it.

#### chisigma

##### Well-known member
If X is a continuous random variable and g is a continuous function
defined on X (Ω), then Y = g(X ) is a continuous random variable.
Prove or disprove it.
If X is a continous r.v. then $F_{X} (x) = P \{ X < x\}$ is continous. Now if $y=g(x)$ is continous then $x=g^{-1} (y)$ is also continous and the same is for...

$$F_{Y} (y) = P \{g(X) < y\} = P \{X < g^{-1} (Y)\} = F_{X} (g^{-1} (y))$$

Kind regards

$\chi$ $\sigma$