PlanetMath: continuity of natural power

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continuity of natural power

(Theorem)

Theorem 1 Let

be arbitrary positive integer. The power function $x\mapsto x^n$ from $\mathbb{R}$ to $\mathbb{R}$ (or $\mathbb{C}$ to $\mathbb{C}$ ) is continuous in each point

Proof. Let $\varepsilon$ be any positive number. Denote and $x^n-x_0^n = \Delta$ . Then identically

$\displaystyle \Delta = (x-x_0)(x^{n-1}+x^{n-2}x_0+...+x_0^{n-1}).$

Taking the absolute value and using the triangle inequality give

$\displaystyle \vert\Delta\vert = \vert h\vert\cdot\vert x^{n-1}+x^{n-2}x_0+...+... ...h\vert\cdot(\vert x^{n-1}\vert+\vert x^{n-2}x_0\vert+...+\vert x_0^{n-1}\vert).$

But since $\vert x\vert = \vert x_0+h\vert \leqq \vert x_0\vert+\vert h\vert$ and also $\vert x_0\vert \leqq \vert x_0\vert+\vert h\vert$ , so each summand in the parentheses is at most equal to $(\vert x_0\vert+\vert h\vert)^{n-1}$ , and since there are

summands, the sum is at most equal to $n(\vert x_0\vert+\vert h\vert)^{n-1}$ . Thus we get

$\displaystyle \vert\Delta\vert \leqq n\vert h\vert(\vert x_0\vert+\vert h\vert)^{n-1}.$

We may choose $\vert h\vert < 1$ ; this implies

$\displaystyle \vert\Delta\vert \leqq n\vert h\vert(\vert x_0\vert+1)^{n-1}.$

The right hand side of this inequality is less than $\varepsilon$ as soon as we still require

$\displaystyle \vert h\vert < \frac{\varepsilon}{n(\vert x_0\vert+1)^{n-1}}.$

This means that the power function $x\mapsto x^n$ is continuous in the point

Note. Another way to prove the theorem is to use induction on and the rule 2 in limit rules of functions.

"continuity of natural power" is owned by pahio.

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Cross-references: limit rules of functions, induction, inequality, right hand side, implies, sum, triangle inequality, absolute value, point, continuous, power function, integer, positive
There are 2 references to this entry.

This is version 3 of continuity of natural power, born on 2006-02-05, modified 2006-02-05.
Object id is 7590, canonical name is ContinuityOfNaturalPower.
Accessed 1007 times total.

Classification:

AMS MSC:	26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
	26C05 (Real functions :: Polynomials, rational functions :: Polynomials: analytic properties, etc.)

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