Abstract: A probe is made into the continuous random variable function theorem about its advantages and disadvantages, on basis of which, the theorem is promoted, and the new theorem thus produced successfully solves the problem that the original theorem is long blamed for its rigid demand of its functions being a strictly monotonic harsh one and thus helps generalize the theorem to piecewise monotone functions, whereby the range of applications has been greatly broadened, and it finds us a new tool for solving the density function of a continuous random variable function.