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 $\R$.

$\blacksquare$