Math Problem Statement
Iff(x) = { 1−|x| / 1+x, x≠−1 ; 1, x=−1 } where [·] represents the greatest integer function, determine the continuity of f([2x]).
Solution
We are analyzing the function , where is defined as:
\begin{cases} \frac{1 - |x|}{1 + x}, & x \neq -1, \\ 1, & x = -1, \end{cases}$$ and $$[ \cdot ]$$ represents the greatest integer function. We are tasked with determining the continuity of $$f([2x])$$, focusing on specific points. --- ### Solution: 1. **Expression for $$f([2x])$$:** The greatest integer function, $$[2x]$$, outputs the largest integer less than or equal to $$2x$$. $$f([2x])$$ depends on the behavior of $$f(x)$$ at integer values since $$[2x]$$ maps $$2x$$ to an integer. 2. **Behavior of $$f(x)$$:** For $$x \neq -1$$, $$f(x) = \frac{1 - |x|}{1 + x}$$. For specific values of $$x$$, evaluate $$f(x)$$: - If $$x = 0$$: $$f(0) = \frac{1 - |0|}{1 + 0} = 1$$, - If $$x = 1$$: $$f(1) = \frac{1 - |1|}{1 + 1} = 0$$, - If $$x = -1$$: $$f(-1) = 1$$ (from the definition). 3. **Continuity of $$f([2x])$$:** Since $$[2x]$$ takes integer values, consider the continuity of $$f(x)$$ at integers. - At $$x = -1$$: $$f(-1) = 1$$. The left-hand limit ($$x \to -1^-$$) does not equal the right-hand limit ($$x \to -1^+$$) due to the discontinuity in the piecewise definition of $$f(x)$$. Therefore, $$f([2x])$$ is **discontinuous at $$x = -1$$**. - At $$x = 0$$, $$x = \frac{1}{2}$$, and $$x = 1$$: Check the values of $$f([2x])$$. Since the greatest integer function produces consistent results for $$2x$$ near these points, $$f([2x])$$ remains **continuous at these points**. --- ### Final Answer: The correct option is **(1) discontinuous at $$x = -1$$**. Would you like further details or examples? Let me know! --- ### Related Questions: 1. What is the greatest integer function and how does it affect continuity? 2. How does $$f([2x])$$ behave for non-integer values of $$x$$? 3. Why does $$f(x)$$ depend on its piecewise definition for continuity? 4. Can $$f([2x])$$ be discontinuous at other points if $$f(x)$$ is not? 5. How does the derivative of $$f(x)$$ behave around $$x = -1$$? ### Tip: Always check the continuity of piecewise functions by evaluating the left-hand and right-hand limits at critical points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Greatest Integer Function (Floor Function)
Formulas
f(x) = (1 - |x|) / (1 + x) for x ≠ -1; f(x) = 1 for x = -1
[x] = Greatest Integer Function (largest integer ≤ x)
Theorems
Definition of Continuity
Properties of the Greatest Integer Function
Suitable Grade Level
Grades 11-12
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