Math Problem Statement
Solution
The question is asking under what conditions the function is continuous at the point . The options present different mathematical statements, and we need to determine which one correctly describes the condition for continuity.
For a function to be continuous at a point , three conditions must be satisfied:
- The function is defined at , meaning exists.
- The limit as approaches of the function exists, meaning exists.
- The function value and the limit must be equal, meaning .
Now, let's analyze the options:
-
A. : This is only the right-hand limit and does not include the left-hand limit. This is not enough to ensure continuity.
-
B. : This is only the left-hand limit and similarly does not account for the right-hand limit.
-
C. : This does not address continuity in general, but only defines a specific function value at , which is irrelevant for continuity.
-
D. : This statement is the formal definition of continuity at a point because it includes the fact that the limit as from both sides equals the function value at .
-
E. : This would only state that the limit of the function is zero at , which is unrelated to the general condition for continuity.
Thus, the correct answer is D.
Would you like more details on the concept of continuity, or do you have further questions? Here are some related questions:
- What are left-hand and right-hand limits in continuity?
- How do we check the continuity of a function graphically?
- Can a function be continuous at some points and not others?
- What is the difference between continuity and differentiability?
- How does continuity relate to limits in higher dimensions?
Tip: Always remember to check both the function value and the two-sided limit when verifying continuity.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Continuity of Functions
Limits
Formulas
lim x → a f(x) = f(a)
lim x → a+ f(x)
lim x → a− f(x)
Theorems
Continuity at a Point
Suitable Grade Level
Grades 11-12 (or early college calculus)