Math Problem Statement
Determine if the piecewise function h(x) is continuous over all real numbers. If a discontinuity exists, state where it is and the type.
Solution
We are given the piecewise function and need to determine its continuity over all real numbers. The function is defined as follows:
\begin{cases} x - 1, & x < -3 \\ x, & x = -3 \\ x^2 - 13, & -3 < x \leq 0 \\ 13 - x, & x > 0 \end{cases}$$ ### Steps to determine continuity: A function is continuous at a point $$x = a$$ if: 1. $$f(a)$$ exists (the function is defined at $$a$$), 2. $$\lim_{{x \to a^-}} f(x) = \lim_{{x \to a^+}} f(x)$$ (the left-hand limit equals the right-hand limit), 3. $$\lim_{{x \to a}} f(x) = f(a)$$ (the limit equals the value of the function at that point). We'll check for continuity at the key points where the definition of the function changes, namely at $$x = -3$$ and $$x = 0$$. ### 1. Continuity at $$x = -3$$: - The function at $$x = -3$$ is $$h(x) = x$$, so $$h(-3) = -3$$. - Left-hand limit as $$x \to -3^-$$ (from $$h(x) = x - 1$$ when $$x < -3$$): $$\lim_{{x \to -3^-}} h(x) = -3 - 1 = -4$$ - Right-hand limit as $$x \to -3^+$$ (from $$h(x) = x^2 - 13$$ when $$-3 < x \leq 0$$): $$\lim_{{x \to -3^+}} h(x) = (-3)^2 - 13 = 9 - 13 = -4$$ - The left-hand limit equals the right-hand limit, but they do not equal $$h(-3)$$, since $$h(-3) = -3$$. Therefore, **there is a jump discontinuity at $$x = -3$$**. ### 2. Continuity at $$x = 0$$: - The function at $$x = 0$$ is $$h(x) = x^2 - 13$$, so $$h(0) = 0^2 - 13 = -13$$. - Left-hand limit as $$x \to 0^-$$ (from $$h(x) = x^2 - 13$$ when $$-3 < x \leq 0$$): $$\lim_{{x \to 0^-}} h(x) = 0^2 - 13 = -13$$ - Right-hand limit as $$x \to 0^+$$ (from $$h(x) = 13 - x$$ when $$x > 0$$): $$\lim_{{x \to 0^+}} h(x) = 13 - 0 = 13$$ - Since the left-hand limit is $$-13$$ and the right-hand limit is $$13$$, they are not equal. Therefore, **there is a jump discontinuity at $$x = 0$$**. ### Conclusion: The function $$h(x)$$ has two jump discontinuities: - At $$x = -3$$, - At $$x = 0$$. Would you like further clarification on any of these steps or a deeper explanation on continuity? Here are 5 related questions you might consider: 1. What is the definition of a jump discontinuity? 2. How do we check for continuity at an endpoint of an interval? 3. How do different types of discontinuities (e.g., removable, infinite) differ from jump discontinuities? 4. Can a function with a jump discontinuity still be differentiable at that point? 5. How would the continuity change if the definition of $$h(x)$$ were modified at $$x = -3$$ or $$x = 0$$? **Tip**: When analyzing piecewise functions for continuity, always check the boundaries where the function's definition changes, as these are common points of discontinuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Formulas
Left-hand limit
Right-hand limit
Conditions for continuity
Theorems
Continuity at a point
Limit definition
Suitable Grade Level
Grades 11-12