Question

If \(f \in L^{1}(m)\) and \(F(x)=\int_{-\infty}^{x} f(t) d t\), then \(F\) is continuous on \(\mathbb{R}\).

Short answer

The function \(F\) is continuous on \(\mathbb{R}\).

Step-by-step solution

Definition of Integral Function

Given that \(f \in L^{1}(m)\), this implies that \(f\) is Lebesgue integrable over real numbers. Define \(F(x) = \int_{-\infty}^{x} f(t) dt\). Our goal is to show that \(F\) is continuous on \(\mathbb{R}\).

Evaluating Continuity at a Point

To show \(F\) is continuous at any point \(c \in \mathbb{R}\), we need to prove \(\lim_{x \to c} F(x) = F(c)\). Start by considering the expression for \(F(x)\) and \(F(c)\):\[ F(x) = \int_{-\infty}^{x} f(t) dt \] and \[ F(c) = \int_{-\infty}^{c} f(t) dt. \]

Difference of Integrals

Consider the difference \(F(x) - F(c) = \int_{-\infty}^{x} f(t) dt - \int_{-\infty}^{c} f(t) dt\). By properties of definite integrals, this simplifies to:\[ F(x) - F(c) = \int_{c}^{x} f(t) dt. \]

Use of Integrability and Limit

Since \(f\) is integrable, for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(|x - c| < \delta\), the integral \(\left| \int_{c}^{x} f(t) dt \right| < \epsilon\) holds as \(x \to c\). This ensures that \(\lim_{x \to c} \int_{c}^{x} f(t) dt = 0\).

Conclusion of Continuity

Thus, \(\lim_{x \to c} (F(x) - F(c)) = 0\), implying \(\lim_{x \to c} F(x) = F(c)\). Hence, \(F\) is continuous at \(c\) and similarly, it can be shown for any \(c \in \mathbb{R}\). \(F(x) = \int_{-\infty}^{x} f(t) dt\) is continuous on \(\mathbb{R}\).

Key concepts

Continuity

In mathematics, a function is said to be continuous if small changes in the input lead to small changes in the output. This means that you should be able to draw the function without lifting your pencil from the paper. For a function to be continuous at a point, it must satisfy the condition

• limiting value as you approach the point,
• the function value at the point, and
• the two values must be equal.

The exercise we are discussing involves showing that the function \(F(x) = \int_{-\infty}^{x} f(t) dt\) is continuous on \(\mathbb{R}\). In the solution, the goal was to prove that the difference of the integral over a small interval around any point \(c\) tends to zero as the width of the interval approaches zero. As a result, the integral function \(F\) keeps values smoothly without jumps or breaks, reinforcing its continuity.

Real Analysis

Real Analysis is a branch of mathematics that deals with real numbers and real-valued sequences and functions. It is foundational for understanding many deeper mathematical theories. One of the crucial aspects of real analysis is understanding how functions behave over sets of numbers and what can influence their properties, such as continuity and integration.
In our context, we focus on Lebesgue integration, which is a powerful tool in real analysis. It allows us to integrate more complex functions that are not necessarily continuous everywhere. By defining the integral over a set of measure rather than simple intervals, Lebesgue integration captures a broader class of functions and measures their total accumulation over any subset of real numbers.
The exercise demonstrates this by discussing an integrable function \(f\) over real numbers \(\mathbb{R}\), providing insights into how its integral function \(F\) behaves continuously across \(\mathbb{R}\). The ability to analyze this behavior is a fundamental skill in real analysis.

Integral Function

An integral function \(F(x)\) represents the accumulation of a function \(f(t)\)'s values over an interval. Specifically, in the scenario where \(F(x) = \int_{-\infty}^{x} f(t) dt\), \(F\) is defined as the integral of \(f\) from minus infinity up to any particular \(x\). This can be thought of as a "running total" of \(f\)'s values as \(x\) increases.

• It provides a way to study how the sum total of \(f(t)\) evolves.
• Even if \(f\) has complicated behavior like peaks or oscillations, \(F(x)\) reveals its overall trends.
• This kind of function is particularly useful in plotting the growth or shrinking of quantities over time or space.

The original exercise showcases the continuity of \(F\), connecting the properties of \(f\) as a Lebesgue integrable function to ensure \(F(x)\) remains smooth. By scrutinizing these functions, we can gather insights into broader behaviors in mathematical phenomena.