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:
\(\ds \forall x \in \R: \, \) | \(\ds \map {f_{k + 1} } x\) | \(=\) | \(\ds x^{k + 1}\) | Definition of $f_{k + 1}$ | ||||||||||
\(\ds \) | \(=\) | \(\ds x \times x^k\) | Definition of Integer Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {f_1} x \times \map {f_k} x\) | Definition of $f_1$ and $f_k$ |
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 $\R$.
$\blacksquare$