Math Problem Statement
Based on f(x) and g(x), which function is continuous at x = 0?
Solution
To determine which function is continuous at , let's examine the limits and values of and as approaches 0 from both sides.
1. Checking Continuity for at
The function is defined as:
-\frac{1}{2}, & x < 0 \\ 2x, & x \geq 0 \end{cases}$$ To be continuous at $$x = 0$$, $$f(x)$$ must satisfy: $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)$$ - **Left-hand limit** ($$x \to 0^-$$): $$\lim_{x \to 0^-} f(x) = -\frac{1}{2}$$ - **Right-hand limit** ($$x \to 0^+$$): Since $$f(x) = 2x$$ for $$x \geq 0$$, $$\lim_{x \to 0^+} f(x) = 2 \cdot 0 = 0$$ - **Value at $$x = 0$$**: $$f(0) = 2 \cdot 0 = 0$$ Since the left-hand limit ($$-\frac{1}{2}$$) is not equal to the right-hand limit and $$f(0)$$, $$f(x)$$ is **not continuous** at $$x = 0$$. ### 2. Checking Continuity for $$g(x)$$ at $$x = 0$$ The function $$g(x)$$ is defined as: $$g(x) = \begin{cases} -2x, & x < 0 \\ 1, & x \geq 0 \end{cases}$$ To be continuous at $$x = 0$$, $$g(x)$$ must satisfy: $$\lim_{x \to 0^-} g(x) = \lim_{x \to 0^+} g(x) = g(0)$$ - **Left-hand limit** ($$x \to 0^-$$): $$\lim_{x \to 0^-} g(x) = \lim_{x \to 0^-} -2x = -2 \cdot 0 = 0$$ - **Right-hand limit** ($$x \to 0^+$$): Since $$g(x) = 1$$ for $$x \geq 0$$, $$\lim_{x \to 0^+} g(x) = 1$$ - **Value at $$x = 0$$**: $$g(0) = 1$$ Since the left-hand limit ($$0$$) is not equal to the right-hand limit and $$g(0)$$, $$g(x)$$ is **not continuous** at $$x = 0$$. ### Conclusion Neither $$f(x)$$ nor $$g(x)$$ is continuous at $$x = 0$$. Would you like further details or have any questions? Here are some related questions you may find interesting: 1. How would we determine the continuity of a piecewise function for points other than the boundary? 2. What are the general conditions for a function to be continuous at a given point? 3. How does discontinuity affect the integrability of a function? 4. What types of discontinuities exist and how do they differ? 5. Can limits help in analyzing the behavior of functions around discontinuous points? **Tip:** To verify continuity at a point for a piecewise function, ensure that the left-hand limit, right-hand limit, and the function's value at that point are all equal.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Left-hand limit
Right-hand limit
Function evaluation at a point
Theorems
Definition of continuity at a point
Suitable Grade Level
Grades 10-12
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