Math Problem Statement
Исследуя функции на непрерывность
Исследуя функции на непрерывность
Solution
The function defined in your image is a piecewise function:
\begin{cases} x, & x \leq -\pi \\ \sin(x), & -\pi < x < \frac{\pi}{2} \\ 1, & x \geq \frac{\pi}{2} \end{cases}$$ To analyze the continuity of $$f(x)$$, let's check the following conditions at the critical points where the definition of $$f(x)$$ changes: $$x = -\pi$$ and $$x = \frac{\pi}{2}$$. ### 1. Continuity at $$x = -\pi$$: - **Left-hand limit**: When $$x \to -\pi^-$$, $$f(x) = x$$. So, $$\lim_{x \to -\pi^-} f(x) = -\pi$$. - **Right-hand limit**: When $$x \to -\pi^+$$, $$f(x) = \sin(x)$$. So, $$\lim_{x \to -\pi^+} f(x) = \sin(-\pi) = 0$$. - **Value of $$f(-\pi)$$**: From the definition, $$f(-\pi) = -\pi$$. Since the left-hand limit ($$-\pi$$) does not equal the right-hand limit ($$0$$), the function is **not continuous** at $$x = -\pi$$. ### 2. Continuity at $$x = \frac{\pi}{2}$$: - **Left-hand limit**: When $$x \to \frac{\pi}{2}^-$$, $$f(x) = \sin(x)$$. So, $$\lim_{x \to \frac{\pi}{2}^-} f(x) = \sin\left(\frac{\pi}{2}\right) = 1$$. - **Right-hand limit**: When $$x \to \frac{\pi}{2}^+$$, $$f(x) = 1$$. So, $$\lim_{x \to \frac{\pi}{2}^+} f(x) = 1$$. - **Value of $$f(\frac{\pi}{2})$$**: From the definition, $$f(\frac{\pi}{2}) = 1$$. Since the left-hand limit, right-hand limit, and value of the function all equal $$1$$, the function is **continuous** at $$x = \frac{\pi}{2}$$. ### Conclusion: - $$f(x)$$ is **not continuous** at $$x = -\pi$$. - $$f(x)$$ is **continuous** at $$x = \frac{\pi}{2}$$. Would you like a graph or further explanation about continuity or piecewise functions? --- ### 5 Related Questions: 1. What is the definition of continuity for a piecewise function? 2. How can the differentiability of $$f(x)$$ be tested at $$x = -\pi$$ and $$x = \frac{\pi}{2}$$? 3. How is the sine function involved in ensuring smooth transitions in piecewise functions? 4. Could $$f(x)$$ be made continuous at $$x = -\pi$$ with adjustments? 5. What are practical examples of piecewise functions in real-world problems? ### Tip: When analyzing continuity, always check the left-hand and right-hand limits, as well as the function's value at the point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Trigonometry
Formulas
Continuity conditions: left-hand limit, right-hand limit, and function value
Theorems
Definition of continuity at a point
Suitable Grade Level
Grades 10-12
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