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

[bl00d1]

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]

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 continuous r.v. then $F_{X} (x) = P \{ X < x\}$ is continous. Now if $y=g(x)$ is continuous then $x=g^{-1} (y)$ is also continuous and the same is for... $$F_{Y} (y) = P \{g(X) < y\} = P \{X < g^{-1} (Y)\} = F_{X} (g^{-1} (y))$$Kind regards $\chi$ $\sigma$