# A function without a maximum and a minimum

• James LeBron
In summary, a non-continuous function on [0,1] that has no maximum and no minimum is given by f(x) = 1/2 if x is a rational number and x if x is an irrational number. This function breaks up the interval into two parts, with a constant value for rational numbers and a variable value for irrational numbers, making it non-continuous. Additionally, as the function approaches 1, there is always a larger number that can be formed, so there is no maximum.
James LeBron

## Homework Statement

Give an example of a non-continuous function on [0,1] that has no maximum and no minimum.

## Homework Equations

Well, a continuous function on a non-empty compact set will have a maximum and a minimum, so I guess this is why we need a non-continuous function.

## The Attempt at a Solution

Does this work?

f(x) =

1/2 if x is a rational number
x if x is an irrational number.

Does this work? Breaking this up into 1/2s (if rational) and x's (if irrational) hopefully makes it non-continuous, and I suppose there is always some larger number that could be formed as we approach 1 (so no maximum). What do you think?

I would agree.

## 1. What is a function without a maximum and a minimum?

A function without a maximum and a minimum is a type of function that does not have a highest or lowest point, also known as a global maximum or minimum. This means that the function continues infinitely in both the positive and negative directions.

## 2. Can a function without a maximum and a minimum still have local maxima and minima?

Yes, a function without a maximum and a minimum can still have local maxima and minima. Local maxima and minima refer to the highest and lowest points within a specific interval of the function, but they do not represent the overall highest or lowest point.

## 3. How is a function without a maximum and a minimum different from a continuous function?

A function without a maximum and a minimum is different from a continuous function in that a continuous function can have both a maximum and a minimum, while a function without a maximum and a minimum does not have either. Additionally, a continuous function is defined for all values of x, while a function without a maximum and a minimum may not be defined at certain points.

## 4. What is the significance of a function without a maximum and a minimum in real-world applications?

In real-world applications, a function without a maximum and a minimum may represent a situation where there is no absolute highest or lowest point. For example, in a temperature function, there may not be a maximum or minimum temperature that is reached, as the temperature can continue to increase or decrease infinitely.

## 5. Is it possible for a function without a maximum and a minimum to have a limit at a certain point?

Yes, it is possible for a function without a maximum and a minimum to have a limit at a certain point. The limit of the function at that point would be the value that the function approaches as x approaches that point, even though the function does not have a maximum or minimum at that point.

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