Math Problem Statement
Examine the continuity of following function at (1, 1) f(x, y) := ( 1 if x + y ≥ 2, −1 if x + y < 2.
Solution
To examine the continuity of the function at the point , we must check if the following condition holds:
1. Define the function:
The function is defined as:
\begin{cases} 1 & \text{if } x + y \geq 2, \\ -1 & \text{if } x + y < 2. \end{cases}$$ ### 2. Evaluate the function at $$(1, 1)$$: At the point $$(1, 1)$$, we have: $$x + y = 1 + 1 = 2.$$ Since $$x + y \geq 2$$, we use the first case of the function definition. Thus, $$f(1, 1) = 1.$$ ### 3. Compute the limit as $$(x, y) \to (1, 1)$$: We now check the limit of the function as $$(x, y)$$ approaches $$(1, 1)$$. This involves approaching $$(1, 1)$$ from different directions and checking whether the limit is the same. #### Case 1: Approaching along a path where $$x + y \geq 2$$ If we approach $$(1, 1)$$ along a path where $$x + y \geq 2$$, such as along the line $$y = 2 - x$$ for $$x \geq 1$$, the value of the function is: $$f(x, y) = 1.$$ Hence, the limit along this path is: $$\lim_{(x, y) \to (1, 1)} f(x, y) = 1.$$ #### Case 2: Approaching along a path where $$x + y < 2$$ If we approach $$(1, 1)$$ along a path where $$x + y < 2$$, such as along the line $$y = 2 - x$$ for $$x < 1$$, the value of the function is: $$f(x, y) = -1.$$ Hence, the limit along this path is: $$\lim_{(x, y) \to (1, 1)} f(x, y) = -1.$$ ### 4. Conclusion: The limits from different directions do not agree: $$\lim_{(x, y) \to (1, 1)} f(x, y) \neq 1.$$ Thus, the function does not have a limit as $$(x, y) \to (1, 1)$$. Since the limit does not exist, the function is **not continuous** at $$(1, 1)$$. --- Would you like more details on any part of this? Here are 5 related questions to explore further: 1. What are the conditions for a function to be continuous at a point? 2. How can we rigorously compute limits in two variables? 3. What does the difference between one-sided limits tell us about continuity? 4. How do discontinuities impact the behavior of a function near a point? 5. Can a function be continuous along some paths but not others? **Tip:** Always check limits from different directions when dealing with multivariable functions to ensure continuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Continuity
Limits
Formulas
lim_{(x, y) → (1, 1)} f(x, y) = f(1, 1)
f(x, y) = 1 if x + y ≥ 2
f(x, y) = -1 if x + y < 2
Theorems
Definition of Continuity
Two-variable Limit Theorem
Suitable Grade Level
College Level - Multivariable Calculus
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