Question

Suppose \(\mathbf{f}\) has a bounded derivative on \([\mathrm{a}, \mathrm{b}]\), so that \(f^{\prime}(x)\) exists and satisfies an inequality \(\left|f^{\prime}(x)\right|

Short answer

\(f\) has bounded variation with \(V_a^b(f) < C(b-a)\).

Step-by-step solution

Understanding the Problem

We are given a function \(f\) whose derivative \(f'(x)\) exists and is bounded on \([a, b]\) such that \(|f'(x)| < c\). We need to prove that \(f\) is of bounded variation with \( V_a^b(f) < C(b-a) \).

Defining Bounded Variation

A function \(f\) is of bounded variation on \([a, b]\) if there exists a constant \(M\) such that for any partition \(P = \{ x_0, x_1, \ldots, x_n \}\) of \([a, b]\), the variation sum \(\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})| \leq M\).

Applying the Mean Value Theorem

For each subinterval \([x_{i-1}, x_i]\) in the partition, by the Mean Value Theorem, there exists a point \(c_i \in (x_{i-1}, x_i)\) such that \(f(x_i) - f(x_{i-1}) = f'(c_i)(x_i - x_{i-1})\).

Bounding the Variation Sum

Since \(|f'(c_i)| < c\) for all \(c_i\), we have \(|f(x_i) - f(x_{i-1})| = |f'(c_i)||x_i - x_{i-1}| < c(x_i - x_{i-1})\). So, the total variation \(V_a^b(f) = \sum_{i=1}^{n} |f(x_i) - f(x_{i-1})| < c \sum_{i=1}^{n} (x_i - x_{i-1}) = c(b-a)\).

Conclusion

We have shown that the total variation \(V_a^b(f) < c(b-a)\), which confirms that \(f\) is of bounded variation with the bound \(C(b-a)\).

Key concepts

Bounded Derivative

A bounded derivative means that the derivative of a function, denoted as \(f'(x)\), exists and does not exceed a certain limit. In mathematical terms, for every \(x\) in the interval \([a, b]\), the inequality \(|f'(x)| < c\) holds. This constraint implies that the rate at which the function \(f(x)\) changes is controlled and does not become too steep or too flat. When a derivative is bounded, it ensures:

• The function does not have any abrupt spikes.
• The slope of the tangent lines to the function at any point is limited.
• There is a consistent limit to how much the function’s value can change as \(x\) varies over the interval.

This concept is crucial because it helps establish that such a function can be of bounded variation, meaning its overall change is finite and predictable.

Mean Value Theorem

The Mean Value Theorem is a fundamental concept in calculus that provides a link between the derivative of a function and its overall change over an interval. The theorem states that for a continuous function \(f\) on \([a, b]\) that is differentiable over \((a, b)\), there exists at least one point \(c\) in \((a, b)\) such that:\[f'(c) = \frac{f(b) - f(a)}{b - a}\]This equation means there is at least one point where the instantaneous rate of change (given by the derivative) matches the average rate of change over the entire interval. It's particularly useful in proving properties related to bounded variation.

• It assures that any change in the function can be attributed to its derivatives over smaller subintervals.
• The theorem confirms that there is some consistency in how \(f\) behaves when changing from \(x_{i-1}\) to \(x_i\) within a partition.
• This consistency is used to bound the change in \(f\)'s value in those smaller parts of \([a, b]\), contributing to the application of bounds over the whole interval.

This is a key stepping stone for showing that the function \(f\) maintains bounded variation.

Variation Sum

The variation sum of a function is a sum that represents how much a function changes over a specified interval. It involves evaluating partitions of the interval, and computing the sum of absolute differences between function values at consecutive partition points. Mathematically, for a function \(f\) over \([a, b]\), and a partition \(P = \{x_0, x_1, ..., x_n\}\), the variation sum is given by:\[\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})|\]Application of bounded derivatives effectively controls this sum. With each small sub-interval change \(f(x_i) - f(x_{i-1})\) being the product of the derivative at some point \(c_i\) and the length of the sub-interval, as shown through the Mean Value Theorem.

• If the derivative is bounded, the variation sum does not exceed \(c(b-a)\).
• The total of all such bounded changes over \([a, b]\) ensures the total variation is also bounded.
• This supports the conclusion that a bounded derivative leads directly to a bounded total variation of the function \(f\).
• This finite variation sum is what makes \(f\) of bounded variation, restricting the function from wild oscillations.

Thus, understanding the variation sum allows us to establish that a bounded derivative systematically limits the potential variability of the corresponding function over its domain.