Question

show that every polynomial function is every where continuous

Answer 1

The proof that every polynomial function over the real numbers is continuous proceeds in a number of simple steps. It is the combination of these steps that makes it easy and comprehensible. A direct attack on a given polynomial function of possibly high degree and strange constants would drown you in calculation problems very soon. That is in a nutshell the power of mathematics. Find the elementary steps that take you to the target.

So here are your steps:

Prove that constant functions and the identical function are continuous (pretty obvious — write down the definition, basically).
Prove that a product of continuous functions is continuous
Prove that a sum of continuous functions is continuous
Clarify to your self that each polynomial is a sum of functions (called terms) which in turn are products of multiple copies of the identical function x and a constant.

Done.

I Hope This Will help you ✌️