Hello,
A random variable is a measurable application (from a space to another).
A continuous function is measurable *provided you are in a correct sigma-algebra, as Laurent pointed out below ^^.
The composition of two measurable functions is measurable.
Thus the composition of a continuous function and a random variable is a random variable
To be rigorous, you should specify which -algebras you endow the probability spaces with. What always holds true is that measurable functions of random variables are random variables. If, as usual, the spaces are endowed with the Borel -algebra (or its completion), then continuous functions are indeed measurable so your statement is correct.