Math Problem Statement
find discontinuity
Solution
To find the discontinuities of the piecewise function defined as:
\begin{cases} x - 1 & \text{if } x \leq -1, \\ x^2 & \text{if } -1 < x \leq 0, \\ x + 1 & \text{if } x > 0, \end{cases}$$ we need to check the continuity of the function at the points where the definition changes, specifically at $$x = -1$$ and $$x = 0$$. ### 1. Check Continuity at $$x = -1$$: For $$f(x)$$ to be continuous at $$x = -1$$, the following must be true: $$\lim_{x \to -1^-} f(x) = \lim_{x \to -1^+} f(x) = f(-1)$$ **Left-hand limit as $$x$$ approaches $$-1$$ from the left:** $$\lim_{x \to -1^-} f(x) = \lim_{x \leq -1, x \to -1^-} (x - 1) = (-1 - 1) = -2$$ **Right-hand limit as $$x$$ approaches $$-1$$ from the right:** $$\lim_{x \to -1^+} f(x) = \lim_{-1 < x \leq 0, x \to -1^+} (x^2) = (-1)^2 = 1$$ **Function value at $$x = -1$$:** $$f(-1) = (-1) - 1 = -2$$ Since $$\lim_{x \to -1^-} f(x) = -2$$ and $$\lim_{x \to -1^+} f(x) = 1$$, the left-hand and right-hand limits are not equal. Therefore, $$f(x)$$ is **discontinuous** at $$x = -1$$. ### 2. Check Continuity at $$x = 0$$: For $$f(x)$$ to be continuous at $$x = 0$$, the following must be true: $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)$$ **Left-hand limit as $$x$$ approaches $$0$$ from the left:** $$\lim_{x \to 0^-} f(x) = \lim_{-1 < x \leq 0, x \to 0^-} (x^2) = (0)^2 = 0$$ **Right-hand limit as $$x$$ approaches $$0$$ from the right:** $$\lim_{x \to 0^+} f(x) = \lim_{x > 0, x \to 0^+} (x + 1) = 0 + 1 = 1$$ **Function value at $$x = 0$$:** $$f(0) = (0)^2 = 0$$ Since $$\lim_{x \to 0^-} f(x) = 0$$ and $$\lim_{x \to 0^+} f(x) = 1$$, the left-hand and right-hand limits are not equal. Therefore, $$f(x)$$ is **discontinuous** at $$x = 0$$. ### Conclusion The function $$f(x)$$ is discontinuous at: - $$x = -1$$ - $$x = 0$$ These are the points where the function changes its form and fails to satisfy the continuity conditions. Would you like any further details or clarifications? Here are 5 questions related to this topic: 1. How can we determine whether a function is continuous at a given point? 2. What are the types of discontinuities a function can have? 3. How do we find the derivative of a piecewise function? 4. What is the difference between a removable discontinuity and a jump discontinuity? 5. Can a function be continuous but not differentiable at certain points? **Tip:** Always check both the left-hand and right-hand limits when analyzing the continuity of a piecewise function at the boundary points where the function's definition changes.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Functions
Continuity
Limits
Formulas
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Theorems
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Suitable Grade Level
High School
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