Math Problem Statement
Montrer que f est discontinue en tout point de R, où f(x) est défini par : f(x) = 1 si x ∈ Q et f(x) = 0 si x ∉ Q.
Solution
The problem is asking to demonstrate that the function , defined by:
\begin{cases} 1 & \text{if } x \in \mathbb{Q} \\ 0 & \text{if } x \notin \mathbb{Q} \end{cases}$$ is discontinuous at every point of $$\mathbb{R}$$. ### Solution Outline: This function is a classic example known as the **Dirichlet function**, which takes the value 1 for rational numbers and 0 for irrational numbers. We want to show that this function is **discontinuous at every point** of the real line. #### Steps for showing discontinuity: 1. **Definition of continuity**: A function $$f$$ is continuous at a point $$x_0 \in \mathbb{R}$$ if: $$\lim_{x \to x_0} f(x) = f(x_0)$$ That is, the limit of the function as $$x \to x_0$$ must equal the function's value at $$x_0$$. 2. **Behavior of rational and irrational points near any $$x_0$$**: - Around any real number $$x_0 \in \mathbb{R}$$, there are both rational and irrational numbers densely packed. This means for any sequence approaching $$x_0$$, we can find subsequences of both rational numbers (where $$f(x) = 1$$) and irrational numbers (where $$f(x) = 0$$). 3. **Limit at any point $$x_0$$**: - If $$x_0 \in \mathbb{Q}$$, as $$x \to x_0$$, we have: - For a sequence of rational numbers approaching $$x_0$$, $$f(x) = 1$$. - For a sequence of irrational numbers approaching $$x_0$$, $$f(x) = 0$$. - This means the limit doesn't exist because the function takes different values depending on whether the sequence is rational or irrational. The same reasoning applies if $$x_0 \in \mathbb{R} \setminus \mathbb{Q}$$. 4. **Conclusion**: Since the limit does not exist at any point $$x_0 \in \mathbb{R}$$, the function is discontinuous at every point. This proves that the function is **discontinuous at every point of $$\mathbb{R}$$**. --- Would you like a more detailed breakdown of any specific part of this solution? ### 5 related questions: 1. What is the difference between rational and irrational numbers? 2. How do you formally define the continuity of a function at a point? 3. Can a function that changes values like this (Dirichlet function) be continuous anywhere? 4. What are some other famous functions that are discontinuous everywhere? 5. How can we use sequences to prove the existence of a limit? **Tip:** Functions that behave differently on dense sets (like rational and irrational numbers) are often used to construct counterexamples in real analysis!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Real Analysis
Continuity
Discontinuity
Rational Numbers
Irrational Numbers
Formulas
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Theorems
Definition of continuity at a point
Density of rational and irrational numbers
Dirichlet Function
Suitable Grade Level
Grade 12 (Baccalaureate, SM)
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