# Describing Continuous Functions f:X→R | Analysis Problem

• kreil
In summary, we are discussing a "trural" metric on a non-empty set X and exploring what continuous functions f: X -> R would look like in this context. A function f is said to be continuous at a point c in X if the preimage of any open set in R under f is also open in X. With this understanding, we can further investigate the properties of open balls and open sets in this metric. Additionally, if the "trural" metric is actually the discrete metric, then all sets are open and the question becomes simpler.
kreil
Gold Member
Let X be any non-empty set, equipped with the "trural" metric:

$$P_x(x,y)= \{ 0:x=y,1:x \ne y \}$$

Describe all continuous functions f: X -> R (First describe what it means for a function f to be continuous at a point c in X).

I'm really quite lost on this, any help is appreciated.

Josh

What do the open balls in X look like?

I'd never seen the term "trural" metric before. I would call that the "discrete" metric. I would also use
d(x,y) instead of Px(x,y). If x is an argument of the function, what does the subscript x mean?

As AKG suggested, what do open balls look like? More precisely, what do
{y| d(x,y)< 1/2} and {y| d(x,y)< 2} look like? What are the open sets in this metric?

I presume your definition of "continuous function" is one for which f-1(C) is open whenever C is open.

yeah. trural threw me too. if it's the discrete (as it appears, though the notation is odd) all sets are open and the question is easy

## 1. What is a continuous function?

A continuous function is a type of mathematical function that has a smooth and unbroken graph. This means that as the input values of the function change, the output values also change in a continuous and connected manner, without any abrupt jumps or breaks in the graph.

## 2. How is a continuous function defined?

A continuous function is defined as a function that satisfies the following condition: for any small change in the input value, there will be a correspondingly small change in the output value. In other words, as the input value approaches a certain point, the output value will also approach a certain point without any sudden changes.

## 3. How do you determine if a function is continuous?

To determine if a function is continuous, you can use the following three criteria:

• The function must be defined at every point in its domain.
• The limit of the function must exist at every point in its domain.
• The limit of the function at a point must be equal to the value of the function at that point.

## 4. What are some common types of continuous functions?

Some common types of continuous functions include linear functions, quadratic functions, exponential functions, and trigonometric functions. These functions have smooth and unbroken graphs and satisfy the criteria for continuity.

## 5. How are continuous functions used in real life?

Continuous functions are used in various fields of science and engineering, such as physics, economics, and computer science. They are used to model and describe real-world phenomena, make predictions, and solve problems. For example, a continuous function can be used to model the growth of a population, the movement of a car, or the flow of electricity in a circuit.

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