Math Problem Statement
Determine if each function is continuous. If the function is not continuous, find the x-axis location of and classify each discontinuity. The function is f(x)={3,x greater than 2} {x/2 + 1/2, x less than 2}
Solution
To determine if the function is continuous, we need to evaluate the continuity at , since this is the point where the function definition changes. The function is defined as follows:
\begin{cases} 3, & x > 2 \\ \frac{x}{2} + \frac{1}{2}, & x < 2 \end{cases}$$ A function is continuous at a point $$x = a$$ if the following three conditions are met: 1. The function $$f(x)$$ is defined at $$x = a$$. 2. The limit of $$f(x)$$ as $$x$$ approaches $$a$$ exists. 3. The limit of $$f(x)$$ as $$x$$ approaches $$a$$ is equal to $$f(a)$$. ### Step 1: Evaluate $$f(x)$$ at $$x = 2$$ For this function, there is no explicit value given for $$f(2)$$. Since neither part of the function is defined at $$x = 2$$, we cannot directly substitute $$x = 2$$ to find $$f(2)$$. Therefore, $$f(x)$$ is not defined at $$x = 2$$, which means $$f(x)$$ is not continuous at $$x = 2$$. ### Step 2: Evaluate the Left-Hand Limit as $$x \to 2^-$$ To find the left-hand limit as $$x$$ approaches 2 from the left, we use the expression $$\frac{x}{2} + \frac{1}{2}$$: $$\lim_{x \to 2^-} f(x) = \frac{2}{2} + \frac{1}{2} = 1 + 0.5 = 1.5$$ ### Step 3: Evaluate the Right-Hand Limit as $$x \to 2^+$$ To find the right-hand limit as $$x$$ approaches 2 from the right, we use the constant value 3: $$\lim_{x \to 2^+} f(x) = 3$$ ### Step 4: Compare the Limits Since the left-hand limit ($$1.5$$) and the right-hand limit ($$3$$) are not equal, the limit does not exist at $$x = 2$$. Therefore, $$f(x)$$ is not continuous at $$x = 2$$. ### Classification of the Discontinuity Since the left-hand and right-hand limits are not equal, this is a **jump discontinuity** at $$x = 2$$. ### Conclusion The function $$f(x)$$ is **not continuous** at $$x = 2$$, and it has a **jump discontinuity** at that point. Would you like further details or have any questions? --- **Follow-up Questions:** 1. How would the classification change if the left-hand and right-hand limits were equal but did not match $$f(2)$$? 2. What types of discontinuities exist, and how can they be identified? 3. How can we modify $$f(x)$$ to make it continuous at $$x = 2$$? 4. What happens to continuity when we add a point $$f(2) = 3$$ to the function definition? 5. How can we use limits to determine continuity for piecewise functions? **Tip:** Always check both left-hand and right-hand limits when evaluating the continuity of piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Calculus
Continuity
Piecewise Functions
Limits
Discontinuities
Formulas
Limit of a function as x approaches a point
Conditions for continuity at a point
Theorems
Definition of continuity
Jump discontinuity
Suitable Grade Level
Grades 11-12 or College-level Calculus
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