The need may be a function of your instructor.
All polynomials are continuous over their domain (the set of Real numbers). This has already been proven, in a number of ways. If you're allowed to reference such theorems, then I would say that you don't need to provide a proof of continuity. Your function is a 5th-degree polynomial, so it's range is (-∞,+∞).
If you're still not sure, chat up your instructor.
If you'd like to read over some different approaches to proving polynomial continuity, you can google for them. A decent working knowledge of precalculus (limits, sequences, and, of course, proofs in general -- like induction) will assist in comprehending most of the approaches. :cool: