Math Problem Statement
Solution
The question asks about a continuous function on the interval and which of the listed statements is necessarily true. Let us analyze the options:
Key Property of Continuity:
A continuous function ensures that for any , the limit of the function as approaches is equal to the value of the function at :
Analyzing Each Option:
(A) exists on :
This is true for any function, not necessarily just continuous functions. So while this is true, it is not specific to continuity.
(B) If is a maximum of , then :
This is incorrect. A continuous function may have a maximum at an endpoint or a cusp, where the derivative may not exist.
(C) for :
This is a true statement and is the defining property of continuity. It restates the idea that the limit of as equals the value .
(D) for some :
This is not necessarily true. For example, the function is continuous on , but may not be in the interval unless the specific shape of demands it.
(E) The graph of is a straight line:
This is not necessarily true. Continuity does not imply linearity.
Correct Answer:
The correct answer is (C). This is the fundamental property of continuity.
Do you want further clarification or additional examples? Let me know!
5 Related Questions:
- What is the formal definition of continuity at a point ?
- How does differentiability relate to continuity?
- Can a function be continuous but not differentiable? Provide examples.
- How does the Intermediate Value Theorem rely on continuity?
- What is the difference between one-sided continuity and full continuity?
1 Tip:
Always remember: Differentiability implies continuity, but continuity does not imply differentiability.
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Math Problem Analysis
Mathematical Concepts
Continuity of Functions
Limits
Differentiability
Critical Points
Formulas
lim_{x -> x_0} f(x) = f(x_0)
Theorems
Continuity Definition
Maximum and Minimum Values
Suitable Grade Level
Grades 11-12 (or first-year college level calculus)