Math Problem Statement

Sei I ein Intervall in R, das mehr als einen Punkt enthält, und seien f : I → R und 9: I → R stetige Funktionen. Für alle x E In Q sei f(x) = g(x). Beweisen Sie, dass f(x) = g(x) für alle x € I gilt.

Solution

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Math Problem Analysis

Mathematical Concepts

Continuity
Density of Rational Numbers
Real Analysis

Formulas

f(x) = lim_{n → ∞} f(x_n) for x_n → x
g(x) = lim_{n → ∞} g(x_n) for x_n → x

Theorems

The density of rational numbers in real numbers
The continuity property of limits

Suitable Grade Level

Undergraduate (Real Analysis)