Chen's question at Yahoo Answers (continuity of the norm).

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In summary, the conversation discusses a proof about the continuity of a real-valued function N on R^n, given a norm N. The proof involves showing that the difference between N(x) and N(y) is less than or equal to the norm of (x-y), and using this to prove the continuity of N.
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Hello Chen,

First, let's prove that $\left|N(x)-N(y)\right|\leq N(x-y)$ for all $x,y\in\mathbb{R}^n$. We have:

$N(y)=N(x+(y-x))\leq N(x)+N(x-y)\Rightarrow -N(x-y)\leq N(x)-N(y)\qquad (1)$

On the other hand:

$N(x)=N(y+(x-y))\leq N(y)+N(x-y)\Rightarrow N(x)-N(y)\leq N(x-y)\qquad (2)$

From $(1)$ and $(2)$ we clearly deduce that $\left|N(x)-N(y)\right|\leq N(x-y)$.

We'll consider on $\mathbb{R}^n$ the usual distance given by the norm $N$, that is $d(x,y)=N(x-y)$ and the usual distance on $\mathbb{R}$.

Now, fix $x_0\in \mathbb{R}^n$ an let $\epsilon>0$, choosing $\delta=\epsilon$:

$d(x,x_0)<\delta\Rightarrow N(x-x_0)<\delta=\epsilon\Rightarrow \left|N(x)-N(x_0)\right|<\epsilon$

which implies that $N$ is a continuous function with the specified distances.
 

FAQ: Chen's question at Yahoo Answers (continuity of the norm).

What is Chen's question about "continuity of the norm" at Yahoo Answers?

Chen's question is about the concept of continuity in mathematics and how it relates to the norm function, which is a mathematical tool used to measure the size or magnitude of a vector.

Why is this question important in the field of mathematics?

This question is important because continuity is a fundamental concept in mathematics that helps us understand and analyze functions, including the norm function. It is also a key concept in calculus and other advanced mathematical topics.

Can you explain the concept of continuity in mathematics?

Continuity refers to the smoothness or unbrokenness of a function over a given interval. In simpler terms, it means that the graph of a function has no sudden jumps or breaks. A function is considered continuous if it can be drawn without lifting the pencil from the paper.

How does continuity relate to the norm function?

The norm function is continuous because it satisfies the definition of continuity. This means that small changes in the input of the norm function result in small changes in the output. In other words, as the input changes slightly, the output also changes in a smooth and continuous manner.

Are there any real-life applications of continuity and the norm function?

Yes, continuity and the norm function are used in many real-life applications, including physics, engineering, economics, and statistics. For example, the concept of continuity is used to model physical processes and make predictions, while the norm function is used to measure the errors in statistical data and optimize mathematical models.

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