Question

Let \(f: E^{1} \rightarrow E^{1}\) be given by $$ f(x)=\frac{1}{x} \text { if } x \neq 0, \text { and } f(0)=0. $$ Show that \(f\) is bounded on an interval \([a, b]\) iff \(0 \notin[a, b] .\) Is \(f\) bounded on (0,1)\(?\)

Short answer

\(f\) is bounded on \((0, 1)\) because 0 is not in the interval.

Step-by-step solution

Define Bounded Function

A function \( f(x) \) is said to be bounded on an interval \([a, b]\) if there exists a real number \( M \) such that for all \( x \in [a, b] \), \(|f(x)| \leq M\).

Analyze Case 1 where \(0 \notin [a, b]\)

If \( 0 otin [a, b] \), then either \(a > 0\) or \(b < 0\). Since \(f(x) = \frac{1}{x}\) for \(x eq 0\), and \(f\) is continuous and bounded away from zero, \(f(x)\) doesn't approach infinity in this interval. Therefore, \(f\) is bounded on \([a, b]\).

Analyze Case 2 where \(0 \in [a, b]\)

If \( 0 \in [a, b] \), then the function \(f(x) = \frac{1}{x}\) has a discontinuity at \(x = 0\). As \(x\) approaches zero, \(f(x)\) tends towards infinity, indicating that \(f\) is not bounded on \([a, b]\).

Evaluate Boundedness on (0,1)

Examine the interval \((0,1)\). Since \(0\) is not included in the open interval \((0,1)\), \(f(x) = \frac{1}{x}\) is continuous and doesn't reach infinite values in \((0, 1)\). Therefore, \(f\) is bounded on \((0,1)\).

Key concepts

Real Analysis

Real analysis is a branch of mathematics that deals with the study of real numbers and the real-valued functions of a real variable. It particularly focuses on concepts like sequences, series, continuity, and limits. In this context, we examine functions and their behavior over intervals of real numbers, ensuring that we understand their boundedness and continuity.
Real analysis helps in understanding how functions behave over specific domains or intervals. It focuses on defining what it means for a function to be bounded or continuous, which are foundational concepts in calculus. For example, understanding whether a function is bounded on an interval tells us about the existence of limits that the function values approach as we move across this interval.
Through real analysis, we can rigorously define and work through problems involving functions like the one given in the exercise, where a function's behavior near discontinuities or at limits needs careful consideration.

Discontinuity

Discontinuity occurs in a function when there is a sudden break or gap in the graph of the function. This typically happens at points where the function is not defined, or where there is an abrupt change in the function's value. In our example, the function \( f(x) = \frac{1}{x} \) becomes problematic at \( x = 0 \).
When we talk about discontinuity, there are several types, but in the case of our function, the issue is with infinite discontinuity. As \( x \) gets closer to 0, the values of \( f(x) \) tend towards infinity, leading to undefined behavior at exactly \( x = 0 \).
This understanding is crucial when determining if a function is bounded over an interval. Since \( f(x) \) at \( x = 0 \) is undefined and creates a point of discontinuity, \([a, b]\) intervals containing zero cannot be considered bounded because the function does not have a finite value everywhere on the interval.

Continuous Functions

Continuous functions are those that have no breaks, jumps, or discontinuities across their domain. For a function \( f(x) \) to be continuous at a point \( c \), the limit of \( f(x) \) as \( x \) approaches \( c \) should be equal to \( f(c) \).
In our example, for the intervals that do not include zero, like (0,1), the function \( f(x) = \frac{1}{x} \) is continuous throughout the interval because it never hits any undefined points. Continuous functions are generally easier to work with when determining boundedness since their value doesn't suddenly change to infinity or become undefined within the interval of interest.
Understanding continuity is key, especially in calculus and real analysis, because it provides information about the behavior of functions and helps in predicting the limit and behavior at boundary points of intervals.

Intervals in Mathematics

Intervals in mathematics are simply ranges of numbers that denote subsets of the real number line. They are commonly expressed in various forms, such as open or closed intervals, depending on whether or not the endpoints are included.
In our analysis, the interval \([a, b]\) can be closed, which means both \(a\) and \(b\) are included, or open as in \((0,1)\), where 0 and 1 are not part of the interval. Understanding how to work with these different types of intervals allows us to determine properties like boundedness of functions.
Intervals can determine whether a function is bound, especially when considering real-valued functions. If a critical point or discontinuity, like \( x = 0 \) in our function, is excluded from an interval, we can better manage function behavior and evaluate boundedness effectively.