Problem 18 - A Course in Calculus and Real Analysis

Question

Let \(f:[a, \infty) \rightarrow \mathbb{R}\) be a bounded function such that \(f\) is integrable on \([a, x]\) for every \(x \geq a\). Let \(F(x):=\int_{a}^{x} f(t) d t\) for \(x \geq a\). Show that \(F\) is uniformly continuous on \([a, \infty)\).

Step-by-Step Solution

Answer

Given \(f:[a, \infty) \rightarrow \mathbb{R}\) is bounded and integrable, we have \(F(x) = \int_{a}^{x} f(t) dt\). To prove \(F\) is uniformly continuous on \([a, \infty)\), we analyze the difference \(|F(x) - F(y)| = \left|\int_{x}^{y} f(t) dt\right|\) using the boundedness of \(f\). Let \(M\) be the upper bound of \(|f(t)|\) on \([a, \infty)\). We find \(|F(x) - F(y)| \leq M|x - y|\). Choosing \(\delta = \frac{\epsilon}{M}\), we ensure that for \(|x-y| < \delta\), \(|F(x) - F(y)| < \epsilon\). Thus, \(F\) is uniformly continuous on \([a, \infty)\).

Step 1: Write down the definition of uniform continuity

We want to show that the function \(F(x)\) is uniformly continuous on \([a, \infty)\). This means, by the definition of uniform continuity, for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all \(x, y \in [a, \infty)\) where \(|x - y| < \delta\), we have \(|F(x) - F(y)| < \epsilon\). For this, we will now analyze the difference \(|F(x) - F(y)|\) and try to find a suitable \(\delta\).

Step 2: Analyze the difference \(|F(x) - F(y)|\)

We have the following difference to analyze: \[|F(x) - F(y)| = \left|\int_{a}^{x} f(t) dt - \int_{a}^{y} f(t) dt \right|\] By the properties of integrals, we can rewrite this as: \[|F(x) - F(y)| = \left|\int_{x}^{y} f(t) dt\right|\] Now, since the function \(f\) is given as both integrable and bounded on \([a, \infty)\), we can find an upper bound for this difference.

Step 3: Use the boundedness of \(f\) to find an upper bound

Let \(M\) be an upper bound of \(|f(t)|\) for all \(t \in [a, \infty)\). This means \(|f(t)| \leq M\) for every \(t \in [a, \infty)\). We can then write: \[|F(x) - F(y)| = \left|\int_{x}^{y} f(t) dt\right| \leq \int_{x}^{y} |f(t)| dt \leq \int_{x}^{y} M dt = M |x - y|\]

Step 4: Find a suitable \(\delta\) in terms of \(\epsilon\)

Now, we want to find a \(\delta > 0\) such that for any \(x,y \in [a, \infty)\) where \(|x - y| < \delta\), we have \(|F(x) - F(y)| < \epsilon\). Since we already found that \(|F(x) - F(y)| \leq M |x - y|\), we can choose a \(\delta\) as follows: \[\delta = \frac{\epsilon}{M}\]

Step 5: Conclude that \(F\) is uniformly continuous

Now, for any given \(\epsilon > 0\), we have found a suitable \(\delta > 0\) such that for any \(x,y \in [a, \infty)\) where \(|x - y|< \delta = \frac{\epsilon}{M}\), we have \(|F(x) - F(y)| \leq M |x - y| < M \delta = \epsilon\). This proves that the function \(F(x)\) is uniformly continuous on the interval \([a, \infty)\).

Key Concepts

Integral of Bounded Function

Understanding the integral of a bounded function illuminates one of the core principles behind calculus. When we say a function is 'bounded', it means that there is a real number that serves as a limit, which the function cannot exceed or drop below. Imagine a fence that the function is not allowed to jump over or dig under.

When dealing with the integrals of these functions, boundedness ensures that the area under the curve, and consequently the integral, will not spiral towards infinity. For instance, if you're capturing the water flow of a bounded river, you know that the amount of water passing a point (analogous to the integral) will not suddenly become an ocean's worth. It remains contained.

In our exercise, this characteristic allows us to assert a controlled behavior over the integrals of function \(f\), which is necessary for proving the uniform continuity of \(F(x)\). We use the bounded nature of \(f\) to establish limits on how much \(F(x)\) can change within a given interval, setting the stage for the argument that \(F(x)\) is reliably predictable, no matter how far along the x-axis we travel.

Properties of Integrals

Integrals are fundamental to calculus, acting as a tool to measure areas, volumes, and other concepts that add up infinite, infinitesimally small pieces. The properties of integrals are what make them manageable. Understanding these properties is like knowing how to bend and shape a piece of clay; they give us the power to mold expressions into forms we can work with.

For example, one such property is linearity, which tells us that integrals can distribute over addition and that they interact with constant factors in predictable ways. Like a precise recipe, these properties ensure if we carefully follow steps, we can simplify complex expressions into something more digestible.

In our solution, we leverage these properties to simplify the complex-looking expression \(|F(x) - F(y)|\) into a more familiar form, bridging the gap between conceptual understanding and practical application. This approach, making the complex simple, is the essence of problem-solving in mathematics.

Definition of Uniform Continuity

Uniform continuity can be a perplexing concept when first encountered—it demands a function to behave well, not just at one point, but uniformly across its entire domain. To comprehend this, picture a train gliding smoothly along tracks without any abrupt turns—a uniform continuity ensures a smooth ride for every passenger, no matter where they boarded.

Formally, a function \(F\) is uniformly continuous on an interval if for any chosen closeness \(\epsilon > 0\), there's a standard measure of proximity \(\delta > 0\) such that, whenever any two points are within \(\delta\) distance, their function values are within \(\epsilon\) distance. This \(\delta\) doesn't change with location; it's like a universal key, fitting every lock along the tracks.

In our step-by-step solution, we seek such a universal \(\delta\) that could ensure \(F\) performs uniformly. By finding it, as we do when we set \(\delta = \frac{\epsilon}{M}\), we satisfy the definition of uniform continuity and thus ensure that \(F(x)\) will offer no surprises, no matter how far along we might be on the interval \([a, \infty)\).