Math Problem Statement
Solution
This problem requires demonstrating that is differentiable at but not continuous for . Let’s analyze the given function:
Function Definition:
\begin{cases} x^2, & \text{if } x \text{ is rational}, \\ 0, & \text{if } x \text{ is irrational}. \end{cases}$$ --- ### 1. **Analyzing Differentiability at $$x = 0$$:** #### Continuity at $$x = 0$$: For $$f(x)$$ to be differentiable at $$x = 0$$, it must first be continuous at $$x = 0$$. To check this, evaluate the limit of $$f(x)$$ as $$x \to 0$$. - If $$x$$ is rational, $$f(x) = x^2$$, and $$\lim_{x \to 0} x^2 = 0$$. - If $$x$$ is irrational, $$f(x) = 0$$, and $$\lim_{x \to 0} 0 = 0$$. Thus, regardless of whether $$x$$ is rational or irrational, $$\lim_{x \to 0} f(x) = 0$$, which equals $$f(0) = 0$$. Therefore, $$f(x)$$ is continuous at $$x = 0$$. #### Derivative at $$x = 0$$: The derivative is defined as: $$f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h}.$$ Since $$f(0) = 0$$: $$f'(0) = \lim_{h \to 0} \frac{f(h)}{h}.$$ Now consider the two cases for $$f(h)$$: - If $$h$$ is rational, $$f(h) = h^2$$, so $$\frac{f(h)}{h} = h$$, and $$\lim_{h \to 0} h = 0$$. - If $$h$$ is irrational, $$f(h) = 0$$, so $$\frac{f(h)}{h} = 0$$, and $$\lim_{h \to 0} 0 = 0$$. Thus, regardless of whether $$h$$ is rational or irrational, $$\lim_{h \to 0} \frac{f(h)}{h} = 0$$. Therefore, $$f'(0) = 0$$, and $$f(x)$$ is differentiable at $$x = 0$$. --- ### 2. **Analyzing Continuity for $$x \neq 0$$:** To check continuity for $$x \neq 0$$, evaluate the left-hand and right-hand limits of $$f(x)$$ as $$x \to c \neq 0$$. Notice that: - If $$x$$ approaches a rational $$c$$, $$f(x) \to c^2$$. - If $$x$$ approaches an irrational $$c$$, $$f(x) \to 0$$. Since the limit depends on whether $$x$$ is rational or irrational, the limit does not exist for any $$x \neq 0$$. Hence, $$f(x)$$ is not continuous for $$x \neq 0$$. --- ### Conclusion: 1. $$f(x)$$ is differentiable at $$x = 0$$. 2. $$f(x)$$ is not continuous for $$x \neq 0$$. Let me know if you'd like a detailed explanation or have further questions! --- ### Expanding Questions: 1. Why is continuity a prerequisite for differentiability? 2. How do discontinuities affect the behavior of piecewise functions? 3. What are the implications of defining functions differently for rational and irrational inputs? 4. Can a function be differentiable at isolated points but discontinuous elsewhere? 5. What mathematical tools help analyze discontinuities in functions? ### Tip: When analyzing piecewise functions, always test limits separately for each case to understand their behavior fully.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Differentiability
Piecewise Functions
Formulas
Derivative definition: f'(x) = lim_{h->0} (f(x+h) - f(x)) / h
Continuity criterion: lim_{x->c} f(x) = f(c)
Theorems
A function is differentiable at a point only if it is continuous at that point.
Limits of piecewise functions depend on the behavior of each segment.
Suitable Grade Level
Undergraduate (Mathematics or Analysis)
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