continuity of natural power
Theorem.
Let be arbitrary positive integer. The power function from to (or to ) is continuous at each point .
Proof. Let be any positive number. Denote and . Then identically
Taking the absolute value and using the triangle inequality give
But since and also , so each summand in the parentheses is at most equal to , and since there are summands, the sum is at most equal to . Thus we get
We may choose ; this implies
The right hand side of this inequality is less than as soon as we still require
This means that the power function is continuous at the point .
Note. Another way to prove the theorem is to use induction on and the rule 2 in limit rules of functions.
Title | continuity of natural power |
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Canonical name | ContinuityOfNaturalPower |
Date of creation | 2013-03-22 15:39:25 |
Last modified on | 2013-03-22 15:39:25 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 7 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 26C05 |
Classification | msc 26A15 |
Related topic | Exponentiation2 |