Question

If \(f:[a, b] \rightarrow \mathbb{R}\), consider the graph of \(f\) as a subset of \(\mathrm{C}\), namely, \(\\{t+i f(t)\) : \(t \in[a, b]\\}\). The length \(L\) of this graph is by definition the supremum of the lengths of all inscribed polygons. (An "inscribed polygon" is the union of the line segments joining \(t_{j-1}+i f\left(t_{j-1}\right)\) to \(t_{j}+i f\left(t_{j}\right), 1 \leq j \leq n\), where \(\left.a=t_{0}<\cdots

Short answer

Total variation equals length for the graph of \(f\); if \(f\) is absolutely continuous, compute the length with \(\int_a^b \sqrt{1 + (f'(t))^2} \, dt\).

Step-by-step solution

Understanding the Problem

To find the length of the graph of the function \(f:[a, b] \rightarrow \mathbb{R}\) seen as a curve in the complex plane, we consider this curve as a subset of complex numbers \( \{ t + i f(t) : t \in [a, b] \} \). The length \(L\) is defined as the supremum of lengths of all polygons inscribed in this curve.

Define Function in Complex Form

The function is given by \( F(t) = t + i f(t) \), where it maps real numbers from \([a,b]\) to complex numbers on a path in the complex plane.

Verify Total Variation Interpretation

The statement in part (a) says that the length \(L\) is equal to the total variation of the function \(F\) over the interval \([a, b]\). In real terms, this means calculating the supreme sum of the absolute increments of \(F(t)\) over partitions of the interval.

Consider Absolute Continuity for Integration

For part (b), where \(f\) is absolutely continuous, we need to use the arc-length formula for curves. The integral \(L = \int_{a}^{b} \sqrt{1 + (f'(t))^2} \, dt \) gives the length by summing infinitesimally small segments of the curve.

Calculate Length Using Integral

To find \(L\), compute the integral \( \int_{a}^{b} \sqrt{1 + (f'(t))^2} \, dt \), which accounts for the length of the curve by adding the infinitesimal distances derived from the Pythagorean theorem applied to each differential segment.

Key concepts

Total Variation

Total variation is a fundamental concept in real analysis. It measures how much a function changes as you move along its domain. Imagine you are on a hike, tracing a mountain path. Total variation would tell you the cumulative "up and down" distance you traveled, not just the straight-line distance from start to end.
In the context of the given exercise, we have a continuous function on an interval \( [a, b] \), transformed into a path in the complex plane, using \( F(t) = t + i f(t) \). The length of this path, denoted as \( L \), is the total variation of \( F \) over the interval \[ a, b \].
This is calculated by taking the supremum (or the upper limit) of the lengths of polygons that can be inscribed into the path. Essentially, we piece together the changes in \( F(t) \) across each segment of a partitioned interval to understand how wildly \( F \) varies.

• If the total variation is large, the path winds and twists significantly.
• If it’s small, the path is more straightforward.

Absolutely Continuous Function

Absolutely continuous functions are a specific type of continuous functions with the added property of preserving length. This means that not only are these functions smooth and without jumps, but they also transform intervals in a way that respects their measure.
In the problem scenario, if \( f \) is absolutely continuous on \( [a, b] \), we can calculate the arc length \( L \) of the curve using the integral formula:
\[L = \int_{a}^{b} \sqrt{1 + (f'(t))^2} \, dt\]
This formula leverages calculus to sum the tiny, straight segments of the curve, accounting for its overall shape without explicitly measuring every twist and turn.
Important aspects of absolute continuity include:

• Lippchitz Property: If a function \( f \) has bounded derivatives, it satisfies a Lippchitz condition, a stronger form of continuity.
• Relationship with Total Variation: When \( f \) is absolutely continuous, its total variation equates to the arc length obtained from integrating its derivative – this links back to how smoothly \( f \) transitions over the interval.

Complex Plane

The complex plane is a two-dimensional space where each point represents a complex number. It can be visualized like a traditional Cartesian coordinate plane, with the x-axis taking real number values and the y-axis representing imaginary components.
In our exercise, we consider a real-valued function \( f \) transformed into the complex plane with \( t + i f(t) \). Here, each input \( t \) is plotted as a real point, while its output \( f(t) \) becomes the imaginary component. Together, they form points on a curve in the complex plane.
Key features of the complex plane include:

• Visualizing Functions: Functions defined in real terms can be understood more dynamically by plotting them in this plane.
• Complex Numbers: Each point on the plane denotes a complex number, characterized by real and imaginary parts.
• Applications: The complex plane is foundational in fields ranging from electromagnetism to signal processing, showcasing its versatility.

By representing functions as paths or curves in this space, we can leverage mathematical tools to determine properties like arc length, as showcased in this example.