Answer
$ f(x)$ is continuous on $ R $.
Work Step by Step
Since $ x^4+1\gt 0$ for all $ x\in R $, then the domain of the function $ f(x)=(x^4+1)^{3/2}$ is $ R $, and since $ x^4+1$ is a polynomial then it is continuous. Now by using the continuity laws, $ f(x)$ is continuous on $ R $.
