Real Power Function for Positive Integer Power is Continuous

Theorem
Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $f_n: \R \to \R$ be the real function defined as:


 * $\forall x \in \R: \map {f_n} x = x^n$

Then $f_n$ is continuous on $\R$.

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
 * $\forall x \in \R: f_n$ is continuous on $\R$.

$\map P 0$ is the case:
 * $\forall x \in \R: \map {f_0} x = x^0 = 1$

Thus it is seen that $f_0$ is the constant mapping.

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:
 * $\forall x \in \R: \map {f_1} x = x^1 = x$

It follows from Linear Function is Continuous that $f_1$ is continuous on $\R$.

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:
 * $\forall x \in \R: f_k$ is continuous on $\R$.

from which it is to be shown that:
 * $\forall x \in \R: f_{k + 1}$ is continuous on $\R$.

Induction Step
This is the induction step:

From the basis for the induction:
 * $f_1$ is continuous on $\R$.

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

It follows from the Product Rule for Continuous Real Functions that $f_{k + 1}$ is continuous on $\R$.

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 $\R$.