Polynomial is Continuous

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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.

$\blacksquare$


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