Question

In the following cases show that \(V_{f}[I]=+\infty,\) though \(f\) is bounded on I. (In case (iii), \(f\) is continuous, and in case (iv), it is even differentiable on \(I .)\) $$ \text { (i) For } I=[a, b](a

Short answer

Each case leverages pathological behaviors or oscillations to achieve infinite total variation.

Step-by-step solution

Understanding Variation and Total Variation

First, let's understand what it means for a function to have infinite total variation on an interval. The total variation of a function \( f \) on an interval \( I \) is defined as the supremum of the sums of absolute differences \( \sum |f(x_{i+1}) - f(x_i)| \) over all partitions \( P \) of \( I \). We need to show this quantity is infinite despite \( f \) being bounded.

Case (i): Function Definition on Rationals and Irrationals

Consider \( f(x) = 1 \) if \( x \) is rational and \( f(x) = 0 \) if \( x \) is irrational. Given any \( m \), we can create a partition \( P \) of \( I = [a, b] \) such that between each successive partition point there exists both a rational and an irrational number, resulting in \( |\Delta_i f| = 1 \). Thus, the sum of these differences \( S(f, P) = m \), making \( Var[f, I] = +\infty \).

Case (ii): Sinusoidal Oscillation

For \( f(x) = \sin \frac{1}{x} \) over \( I = [a, b] \) with \( f(0) = 0 \), the function oscillates infinitely as \( x \to 0 \). For any partition, there will be subintervals where \( \sin \frac{1}{x} \) varies rapidly, leading to large cumulative sums and thus infinite total variation, as \( x \to 0 \) dominates the bound.

Case (iii): Sum of Variations Over Partition

For \( f(x) = x \cdot \sin \frac{\pi}{2x} \) with \( I = [0,1] \), using the partition \( P_m = \left\{0, \frac{1}{m}, \frac{1}{m-1}, \ldots, \frac{1}{2}, 1\right\} \), we note that \( |f\left(\frac{1}{k}\right) - f\left(\frac{1}{k+1}\right)| \approx \frac{1}{k} \). This forms a harmonic series \( \sum \frac{1}{k} \) which diverges as \( m \to \infty \), proving infinite total variation.

Case (iv): Derivative-Induced Oscillation

For \( f(x) = x^2 \cdot \sin \frac{1}{x^2} \) with \( I = [0,1] \), the function is differentiable and its derivative constructs a similar oscillatory pattern as \( x \to 0 \). As with the previous cases, for any partition, variations at small \( x \) become large enough to cause the total variation to diverge, proving \( Var[f, I] = +\infty \).

Key concepts

Bounded Functions

In mathematics, bounded functions are those that have a limit on their range. This means the function will not produce any values beyond a certain minimum or maximum on a given interval. Regardless of the input values, the results will always stay within these bounds.
For example, if a function is given as bounded by 0 and 1, it means that no matter what value you plug into the function, the output will fall within this range. This property is crucial, especially when assessing whether a function might have infinite total variation.
It's important to note that a function can be bounded yet possess infinite total variation. Total variation refers to the overall amount by which a function's output changes across an interval. If this amount is not limited, even a bounded function can have infinite total variation, leading to interesting characteristics in mathematical analysis.

Rational and Irrational Numbers

Numbers are classified as either rational or irrational. Rational numbers, such as 1/2 or 3, can be expressed as a fraction of two integers. Irrational numbers, like π or √2, cannot be accurately written as a simple fraction.
In mathematical exercises, distinguishing between these types of numbers is often crucial. One scenario involves the function that assigns 1 to rational numbers and 0 to irrational ones within a given interval. This type of function can take on values that oscillate wildly over any partition, contributing to infinite total variation.
The infinite presence of both types of numbers in any interval makes it possible to create partitions where these variations are maximized. This concept is particularly relevant in exercises analyzing the behavior of piecewise functions or those involving limits. Understanding the properties of rational and irrational numbers helps in predicting and explaining the behavior of complex functions.

Sinusoidal Functions

Sinusoidal functions, taken from trigonometry, are functions that involve the sine or cosine of an angle. Their characteristic wave-like shape is useful in modeling periodic or oscillating behavior, such as sound waves or seasonal temperature variations.
For example, consider the function \( f(x) = \sin \left( \frac{1}{x} \right) \). As \( x \) approaches zero, the frequency of oscillation increases dramatically, leading the sine function's graph to resemble a dense collection of waves.
In mathematical analysis, especially when considering infinite total variation, sinusoidal functions showcase a rapidly changing behavior over small intervals, making it challenging to measure precise changes. This infinite oscillation contributes to a function having an infinite total variation, even if it remains bounded.

Partition of Intervals

Partitioning an interval involves dividing it into a series of smaller subintervals. This technique is often used in calculus and analysis to better understand the behavior of functions over a given range.
A well-chosen partition helps to piece out irregularities in a function, assisting with calculations such as the Riemann sum or total variation. By segmenting the interval, one can focus on specific features and how they contribute to the function's overall behavior.
The effectiveness of partitions is illustrated in the exercise through cases where infinite smaller intervals contain rapidly changing functions. Such partitions reveal how some bounded functions still exhibit an infinite total variation by accruing large sums of difference values over their sub-parts. This understanding aids in visualizing more complex mathematical phenomena and is essential for detailed analysis of function behaviors.