Math Problem Statement
Translate to english and solve
It is known that the function \( f \) is defined on the entire real line, continuous at the point \( 0 \), and \( f(0) = 0 \). What can be said about the continuity of the function
\[
g(x) =
\begin{cases}
f(x) + 1, & x \leq 0, \\
f(x) - 1, & x > 0,
\end{cases}
\]
at the point \( x = 0 \)?
Solution
Translation of the Question
It is known that the function is defined on the entire real line, is continuous at the point , and .
What can be said about the continuity of the function
\begin{cases} f(x) + 1, & x \leq 0, \\ f(x) - 1, & x > 0, \end{cases}$$ at the point $$x = 0$$? Choose one: 1. The function $$g$$ is continuous at $$x = 0$$. 2. The function $$g$$ is not continuous at $$x = 0$$. 3. There is not enough information to conclude the continuity of $$g$$ at $$x = 0$$. ### Solution To determine if $$g(x)$$ is continuous at $$x = 0$$, we need to examine the left-hand limit, the right-hand limit, and the value of $$g(0)$$. 1. **Left-hand limit as $$x \to 0^-$$:** $$\lim_{x \to 0^-} g(x) = \lim_{x \to 0^-} (f(x) + 1) = f(0) + 1 = 0 + 1 = 1$$ 2. **Right-hand limit as $$x \to 0^+$$:** $$\lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} (f(x) - 1) = f(0) - 1 = 0 - 1 = -1$$ 3. **Value of $$g(0)$$:** Since $$g(x) = f(x) + 1$$ for $$x \leq 0$$, we have $$g(0) = f(0) + 1 = 0 + 1 = 1$$. The left-hand limit and right-hand limit at $$x = 0$$ are not equal (left-hand limit is 1, and the right-hand limit is -1), so $$g(x)$$ is not continuous at $$x = 0$$. ### Conclusion The correct answer is: - **The function $$g$$ is not continuous at $$x = 0$$.** Would you like more details or have any questions? --- Here are 5 related questions that build on this concept: 1. How would the continuity of $$g(x)$$ change if $$f(x)$$ were not continuous at $$x = 0$$? 2. What is the significance of left-hand and right-hand limits in determining continuity? 3. If $$g(x)$$ were defined differently for $$x > 0$$, could it still be continuous at $$x = 0$$? 4. How does the continuity of $$f(x)$$ at $$x = 0$$ affect the overall behavior of $$g(x)$$? 5. What are some real-life applications of piecewise functions like $$g(x)$$? **Tip:** When checking continuity at a point, always compare the left-hand limit, right-hand limit, and the function's value at that point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Left-hand limit \( \lim_{x \to 0^-} g(x) \)
Right-hand limit \( \lim_{x \to 0^+} g(x) \)
Theorems
Definition of Continuity at a Point
Suitable Grade Level
High School (Grades 11-12)