Math Problem Statement
- For what values of x is g continuous? (multiple choice question) g(x) = { 0 if x is rational x if x is irrational x = 0 x = −1 x = 1 x is in double-struck R none of these
Solution
The function is defined as:
\begin{cases} 0 & \text{if } x \text{ is rational} \\ x & \text{if } x \text{ is irrational} \end{cases}$$ ### Continuity Condition A function $$g(x)$$ is continuous at a point $$x_0$$ if: $$\lim_{x \to x_0} g(x) = g(x_0)$$ For this specific function, we need to check if the limit of $$g(x)$$ as $$x$$ approaches a certain point $$x_0$$ (both for rational and irrational numbers) is equal to $$g(x_0)$$. ### Evaluating at specific points 1. **At $$x = 0$$**: - If $$x$$ is rational, $$g(x) = 0$$. - If $$x$$ is irrational, $$g(x) = 0$$. Therefore, $$g(x)$$ is continuous at $$x = 0$$. 2. **At other points (e.g., $$x = 1$$, $$x = -1$$, or any other real number)**: - If $$x_0 \neq 0$$, the function is discontinuous because the values differ depending on whether $$x$$ is rational or irrational: - If $$x$$ is rational, $$g(x) = 0$$. - If $$x$$ is irrational, $$g(x) = x_0$$. The limit will not exist because the rational and irrational values do not approach the same value at these points. Hence, $$g(x)$$ is **discontinuous** at all points except $$x = 0$$. ### Conclusion The function is continuous **only at $$x = 0$$**. Therefore, the correct answer is: $$\boxed{x = 0}$$ Would you like more details or have any questions? Here are 5 related questions to further explore this topic: 1. What does it mean for a function to be continuous on the real line? 2. How do you formally prove that a function is continuous at a point? 3. What are examples of functions that are continuous everywhere except at a single point? 4. How can we apply the concept of limits to analyze continuity for piecewise functions? 5. What are the characteristics of functions that behave differently for rational and irrational numbers? **Tip:** Always check the behavior of a function near a point from both the rational and irrational directions when analyzing continuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Real Analysis
Formulas
Continuity condition: lim(x -> x0) g(x) = g(x0)
Theorems
Continuity of piecewise functions
Limit behavior for rational and irrational numbers
Suitable Grade Level
Grades 11-12, College Level