Examples of Functions in Real Analysis
In real analysis, a function is a fundamental object of study, and various properties of functions are crucial for understanding their behavior. One of the important concepts in this field is the idea of a function that is continuous, bounded, and attains its minimum but does not attain its maximum. We will explore some examples of such functions by delving into their properties and behavior.
Example 1: (f(x) frac{1}{x - 1}) for (x in (1, 2))
The function (f(x) frac{1}{x - 1}) is a prime example of a function that meets these criteria. Let's examine its properties:
Continuity
The function is continuous on the interval ((1, 2)) because it is defined for all (x) in that interval and has no discontinuities. This means that for any (x_0) in ((1, 2)), (f(x)) is defined and the limit of (f(x)) as (x) approaches (x_0) is equal to (f(x_0)).
Boundedness
The function (f(x) frac{1}{x - 1}) is bounded on ( (1, 2)). As (x) approaches 1 from the right, (f(x)) grows without bound and approaches infinity. Conversely, as (x) approaches 2, (f(x)) approaches (frac{1}{1} 1). Therefore, the function is bounded above by 1.
Minimum
The minimum value of the function occurs as (x) approaches 2, where (f(2)) approaches 1. However, since the function is defined on the open interval ((1, 2)), it does not actually attain the value 1 at (x 2). The infimum, or the greatest lower bound, of (f(x)) in ((1, 2)) is 1, which is the limit as (x) approaches 2 from the left, denoted as (lim_{x to 2^-} f(x) 1).
Maximum
The function does not attain a maximum value on this interval because as (x) approaches 1 from the right, (f(x)) approaches infinity. Therefore, there is no point in the interval where (f(x)) reaches a maximum value.
Example 2: (f(x) (x - frac{3}{2})^2) for (x in (1, 2))
Another example is given by the function (f(x) (x - frac{3}{2})^2). This function is also continuous and attains its minimum value of 0 at (x frac{3}{2}). It is bounded with values lying between 0 and (frac{1}{4}) but does not attain the value of (frac{1}{4}) for any (x) in the interval (1, 2).
Properties:
1. Continuity: The function is continuous on ((1, 2)), as it is a polynomial function, which is continuous everywhere.
2. Minimum: The minimum value is 0, which occurs at (x frac{3}{2}).
3. Boundedness: The function is bounded, with values ranging from 0 to (frac{1}{4}).
Generalization
For any constant (c in (1, 2)), the function (f(x) x - c) also fits this description. This is a linear function that is continuous and bounded, and it attains its minimum value, which is (-c), at (x 0). However, it does not attain a maximum value on the interval ((1, 2)).
Example 3: (f(x) frac{x - 1}{x - 2})
Let's consider another function, (f(x) frac{x - 1}{x - 2}). This function is designed to achieve its minimum somewhere in the interior of the interval ((1, 2)) and its maximum on the boundary. When restricted to ((1, 2)), the function retains its minimum but loses its maximum value. This function is continuous on the interval ((1, 2)) and attains its minimum value, which is -1, at (x 1.5).
Properties:
1. Continuity: The function is continuous on ((1, 2)).
2. Minimum: The minimum value is -1, which occurs at (x 1.5).
Example 4: (f(x) cos(x))
The function (f(x) cos(x)) is also relevant in this context. The cosine function oscillates between -1 and 1, and on the interval ((1, 2)), it attains its maximum and minimum values but is symmetric around the origin. When restricted to ((1, 2)), it oscillates but does not reach its maximum or minimum value at the boundaries.
Conclusion
These examples illustrate the properties of functions that are continuous, bounded, attain their minimum, but do not attain their maximum on a specific interval. They are valuable in understanding the behavior of functions in real analysis and provide insights into mathematical concepts such as continuity, boundedness, and the existence of extrema.