Math Problem Statement
Let f(x) = x for rational numbers and f(x) = 0 for irrational numbers. Prove that f is continuous at x = 0.
Solution
To prove that the function , defined as
\begin{cases} x & \text{if } x \text{ is rational} \\ 0 & \text{if } x \text{ is irrational} \end{cases}$$ is continuous at $$x = 0$$, we need to show that: $$\lim_{x \to 0} f(x) = f(0).$$ ### Step 1: Evaluate $$f(0)$$ First, we calculate $$f(0)$$. Since 0 is a rational number, we use the definition of $$f(x)$$ for rational numbers: $$f(0) = 0.$$ ### Step 2: Check the limit $$\lim_{x \to 0} f(x)$$ Now, let's compute $$\lim_{x \to 0} f(x)$$. We need to check the behavior of $$f(x)$$ as $$x$$ approaches 0, regardless of whether $$x$$ is rational or irrational. - If $$x$$ is **rational**, then $$f(x) = x$$. - If $$x$$ is **irrational**, then $$f(x) = 0$$. Thus, for any sequence of rational numbers $$x_n \to 0$$, we have $$f(x_n) = x_n \to 0$$. Similarly, for any sequence of irrational numbers $$x_n \to 0$$, we have $$f(x_n) = 0$$. In either case, the limit of $$f(x)$$ as $$x \to 0$$ is 0. Thus, we conclude: $$\lim_{x \to 0} f(x) = 0.$$ ### Step 3: Conclude continuity at $$x = 0$$ Since $$\lim_{x \to 0} f(x) = f(0) = 0$$, we can conclude that $$f(x)$$ is continuous at $$x = 0$$. ### Summary The function $$f(x)$$ is continuous at $$x = 0$$ because the limit of $$f(x)$$ as $$x \to 0$$ equals $$f(0)$$, and both are equal to 0. --- Would you like more details, or do you have any questions? Here are five related questions to explore: 1. Is $$f(x)$$ continuous at any other points on the real line? 2. What is the behavior of $$f(x)$$ at any non-zero rational number? 3. Can you describe a function that is continuous at all rational points but discontinuous at irrational points? 4. How does this function compare with the Dirichlet function, which is $$1$$ for rationals and $$0$$ for irrationals? 5. How does one use the concept of sequential limits to prove continuity? **Tip:** A function's continuity depends not only on its value at a point but also on how it behaves as inputs approach that point!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Real Analysis
Continuity
Limits
Irrational and Rational Numbers
Formulas
\lim_{x \to 0} f(x) = f(0)
Theorems
Limit Definition of Continuity
Sequential Criterion for Limits
Suitable Grade Level
University Level (Advanced Calculus or Real Analysis)
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