# Polynomial is Continuous

From ProofWiki

## Theorem

A (real) polynomial function is continuous at every point.

Thus a (real) 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 Product Rule for Limits of Functions shows us that:

- $\displaystyle \forall k \in \N: \lim_{x \to c} \ x^k = c^k$

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 Combined Sum Rule for Limits of Functions, 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.

$\blacksquare$

## Sources

- K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*(1977)... (previous)... (next): $\S 8.13$ - K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*(1977)... (previous)... (next): $\S 9.2$