Real Polynomial Function 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 1

Let $f_n$ be an arbitrary real polynomial function of degree $n$.

The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:

$f_n$ is continuous on $\R$.


$\map P 0$ is the case $f_0$, where $f_0$ is of zero degree.

Such a real polynomial function is a constant function.

It follows from Constant Real Function is Continuous that $f_0$ is continuous on $\R$.

Thus $\map P 0$ is seen to hold.


Basis for the Induction

$\map P 1$ is the case:

$f_1$ is continuous on $\R$.

Such a real polynomial function is a linear function.

Thus $\map P 1$ is seen to hold.


This is the basis for the induction.


Induction Hypothesis

Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.


So this is the induction hypothesis:

An arbitrary real polynomial function of degree $k$ is continuous on $\R$


from which it is to be shown that:

An arbitrary real polynomial function of degree $k + 1$ is continuous on $\R$.


Induction Step

This is the induction step:

$\forall x \in \R: \map {f_{k + 1} } x = a_{k + 1} x^{k + 1} + \map {f_k} x$

for some real polynomial function of degree $k$, where $a_{k + 1} \in \R$


From the induction hypothesis $f_k$ is continuous on $\R$.

From Real Power Function for Positive Integer Power is Continuous‎, the real function $x \mapsto x^{k + 1}$ is continuous.

Hence from the Combined Sum Rule for Continuous Real Functions:

$f_{k + 1}$ is continuous.


So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall n \in \Z_{\ge 0}: f_n$ is continuous on $\R$.

$\blacksquare$


Proof 2

From Linear Function is Continuous‎, setting $\alpha = 1$ and $\beta = 0$, we have that:

$\ds \lim_{x \mathop \to c} x = c$

Repeated application of the Product Rule for Limits of Real Functions shows us that:

$\ds \forall k \in \N: \lim_{x \mathop \to c} x^k = c^k$


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

By repeated application of the Combined Sum Rule for Limits of Real Functions, we find that:

$\ds \lim_{x \mathop \to c} \map P x = \map P c$


So whatever value we choose for $c$, we have that $\map P x$ is continuous at $c$.


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

$\blacksquare$


Sources

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