Problem 10 - Basic Analysis: Introduction to Real Analysis

Question

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) be continuous functions. Suppose that for all rational numbers \(r, f(r)=g(r)\). Show that \(f(x)=g(x)\) for all \(x\).

Step-by-Step Solution

Answer

Since both functions are continuous and equal on dense rationals, they are equal on all reals.

Step 1: Understand Continuity

Both functions \( f \) and \( g \) are continuous on the real number line \( \mathbb{R} \). Continuity means that for any point \( x \) in the domain, the limit of \( f \) as it approaches \( x \) equals \( f(x) \), and similarly for \( g \).

Step 2: Equal Values on Rational Numbers

We are given that for all rational numbers \( r \), \( f(r) = g(r) \). Recall that the set of rational numbers \( \mathbb{Q} \) is dense in \( \mathbb{R} \), meaning that in any interval of \( \mathbb{R} \), you can find a rational number.

Step 3: Denseness and Limit Points

Given that the rationals are dense in \( \mathbb{R} \), for any real number \( x \), there exists a sequence of rational numbers \( \{r_n\} \) converging to \( x \). Because \( f \) and \( g \) are continuous, \( \lim_{n \to \infty} f(r_n) = f(x) \) and \( \lim_{n \to \infty} g(r_n) = g(x) \).

Step 4: Conclude Equality Using Limits

Since \( f(r_n) = g(r_n) \) for every rational \( r_n \) in the sequence, by continuity of \( f \) and \( g \), the limits must also be equal. Thus, \( f(x) = g(x) \) for all real numbers \( x \).

Key Concepts

real numbers

Real numbers are an expansive set that includes all the numbers we encounter on a continuous number line. They are composed of both rational and irrational numbers, making them incredibly important in mathematics.

**Key Characteristics of Real Numbers:**

• Real numbers can be positive, negative, or zero.
• They encompass all possible magnitudes and can have decimal or fractional expressions.
• The continuity of real numbers makes them a perfect domain for many functions in calculus and analysis.

Mathematically, the real numbers are represented with the symbol \( \mathbb{R} \), and they are infinite and unbounded, stretching from negative infinity to positive infinity. **Density of Real Numbers:** Because there are no gaps between real numbers, they form a complete set, where between any two real numbers, there is another real number. This dense nature effectively supports the continuity of functions like those in calculus.

rational numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Symbolically, rational numbers are represented by the set \( \mathbb{Q} \). Examples include 1/2, -3, and 4.5, which can be written as 9/2.

**Properties of Rational Numbers:**

• They are closed under addition, subtraction, multiplication, and division (except by zero).
• Each rational number can be expressed in its simplest form, leading to an unending or repeating decimal expansion.

**Rational Numbers and Denseness in Real Numbers:** A crucial property is their denseness in real numbers, meaning between any two real numbers, however close they may be, there exists a rational number. This characteristic of rational numbers plays a significant role in the study of limits and continuity in mathematical analysis.

limit points

In mathematics, particularly in the area of analysis, a limit point (or accumulation point) of a sequence is a point that the sequence approaches as it progresses through its terms. It is vital in discussions about convergence and continuity.

**Understanding Limit Points:**

• A sequence converges to a limit point if its terms get arbitrarily close to the point as it progresses.
• A limit point can lie within the set of points generated by the sequence or outside of it.

**Relation to Continuous Functions:** For a function to be continuous, as explored in the exercise, the value of the function at any point, including at the limit points, relies on this idea of convergence. Here, rational numbers' density means any sequence of rationals can have a real number as a limit point, and continuity ensures the function's value is defined and stable at these points.

Thus, the identification with limit points confirms why functions equal at rationals may indeed be equal across all real numbers when continuity is present, as any real number can serve as a limit point of a sequence of rationals.