Real Polynomial Function is Continuous

Theorem
A polynomial function is continuous at every point.

Thus a polynomial function is continuous on every interval of $\R$.

Proof
From Linear Function is Continuous‎, setting $\alpha = 1$ and $\beta = 0$, we have that $\displaystyle \lim_{x \to c} x = c$.

Repeated application of the Combination Theorem for Functions: Product of Limits shows us that $\displaystyle \lim_{x \to c} x^k = c^k$ for all $k \in \N$.

Now let $P \left({x}\right) = a_n x^N + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$.

Now by repeated application of the Combination Theorem for Functions: Sum of Limits, we find that $\displaystyle \lim_{x \to c} P \left({x}\right) = P \left({c}\right)$.

So whatever value we choose for $c$, we have that $P \left({x}\right)$ is continuous at $c$.

From the definition of continuity on an interval, the second assertion follows.