Real Polynomial Function is Continuous

Theorem
A polynomial is continuous at every point.

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

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

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 $$\lim_{x \to c} P \left({x}\right) = P \left({c}\right)$$.

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