For a definition of polynomials, see Definition 1, here. For a definition of continuous functions, see Definition 1, here.
Lemma 1: f(x)=x is continuous
(1) Let ε be any arbitrary value.
(2) Let δ = ε
(3) For any point c, it is clear that if x lies in (c - δ, c + δ), then f(x)=x lies in (f(c) - ε, f(c) + ε )
Corollary 1.1 : Polynomials are continuous
(1) The function f(x)=x is continuous. [See Lemma 1 above]
(2) Since the product of continuous functions is continuous [See Lemma 3, here], then f(x)=xn where n is a positive integer is also continuous.
(3) Since f(x)=C is continuous [See Lemma 1, here], it follows that any function of the form cxn is also continuous.
(4) Since the addition of continuous functions is continuous [See Lemma 2, here], it follows that any polynomial function is continuous since it consists of the form:
f(x) = c0 + c1x + c2x2 + ... + cnxn
where each ci is a constant.
Lemma 2: The Derivative of a polynomial is itself a polynomial
(1) The derivative of each term of a polynomial is itself a term of a polynomial [See the Lemma 2, here]
(2) So, it follows that the derivative itself is also a polynomial. [See Definition 1, here]
Corollary 2.1: The derivative of a polynomial is a continuous function.
This follows directly from Lemma 2 above and Corollary 1.1 above.