# Continuous => limited in region

• Kruger
In summary, to show that a function f:R->R is limited in a suitable environment of x', we can use the inequality |f(a)|=<|f(a)-f(y)|+|f(y)| and choose a suitable d and M to show that for any epsilon greater than 0, there exists a neighborhood of x' where |f(y)|<M for all y in that neighborhood. This follows from the epsilon delta definition of continuity.
Kruger

## Homework Statement

I've to show that if f:R->R is continuous in x' then f is limited in a suitable environment of x'.

2. The attempt at a solution

My lecturer said we should use the following inequality

|f(a)|=<|f(a)-f(y)|+|f(y)|

But how should I go on, I know I have to show something like this:

It exists d>0: it exist c element of [x'-d, x'+d]=I ==> |f(x)|=<|f(c)| for every x in I.

But how should I go on?

Always include relevant definitions. "limited in a suitable environment" is not standard terminology, and I don't know what it means. It sounds like "bounded in some neighborhood", but this would follow imediately from the epsilon delta definition of continuity.

I thought that, too. I mean that it follows imidiately, but our lecturer said that we've to take |f(a)|=<|f(a)-f(y)|+|f(y)| to show that.

With limited in a suitable environment it is ment what you said (bounded in some neightboorhood).

Right, so pick any e>0, then a d>0 so that |y-a|<d => |f(y)-f(a)|<e, and then use that inequality to show there is an M with |f(y)|<M, all y in some neighborhood of a.

## 1. What is the definition of "continuous" in the context of regions?

Continuous refers to a function or set of values that can take on any value within a given range without any gaps or interruptions. In the context of regions, this means that a continuous region has no breaks or discontinuities, and all points within the region are connected.

## 2. How does a continuous region differ from a limited region?

A continuous region has no boundaries or limitations, meaning that it can extend infinitely in all directions. On the other hand, a limited region has defined boundaries and does not extend infinitely. This means that there are points outside of a limited region that cannot be included within it.

## 3. Can a continuous region be bounded?

No, a continuous region cannot be bounded. As mentioned, a continuous region has no boundaries or limitations, so it cannot be contained within a specific area. It extends infinitely in all directions.

## 4. What are some examples of continuous regions?

Some examples of continuous regions include the real number line, a circle, a cylinder, or a plane. These regions have no breaks or interruptions and can extend infinitely in all directions.

## 5. How is the concept of continuity important in mathematics and science?

The concept of continuity is important in mathematics and science because it allows us to model and analyze real-world phenomena. For example, in calculus, the concept of continuity is crucial for understanding and solving problems involving rates of change. In science, continuity is used to describe continuous functions and processes, such as the flow of fluids or the growth of populations. It is also important in the study of limits and derivatives, which are fundamental concepts in many scientific fields.

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