Math Problem Statement

If f is a continuous function on [a, b], which of the following is necessarily true? (Options A to E provided)

Solution

The question asks about a continuous function ff on the interval [a,b][a, b] 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 x0(a,b)x_0 \in (a, b), the limit of the function as xx approaches x0x_0 is equal to the value of the function at x0x_0: limxx0f(x)=f(x0).\lim_{x \to x_0} f(x) = f(x_0).

Analyzing Each Option:

(A) ff exists on (a,b)(a, b):

This is true for any function, not necessarily just continuous functions. So while this is true, it is not specific to continuity.

(B) If f(x0)f(x_0) is a maximum of ff, then f(x0)=0f'(x_0) = 0:

This is incorrect. A continuous function may have a maximum at an endpoint or a cusp, where the derivative f(x0)f'(x_0) may not exist.

(C) limxx0f(x)=f(limxx0x)\lim_{x \to x_0} f(x) = f\left(\lim_{x \to x_0} x\right) for x0(a,b)x_0 \in (a, b):

This is a true statement and is the defining property of continuity. It restates the idea that the limit of f(x)f(x) as xx0x \to x_0 equals the value f(x0)f(x_0).

(D) f(x)=0f'(x) = 0 for some x[a,b]x \in [a, b]:

This is not necessarily true. For example, the function f(x)=x2f(x) = x^2 is continuous on [a,b][a, b], but f(x)f'(x) may not be 00 in the interval unless the specific shape of f(x)f(x) demands it.

(E) The graph of ff 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:

  1. What is the formal definition of continuity at a point x0x_0?
  2. How does differentiability relate to continuity?
  3. Can a function be continuous but not differentiable? Provide examples.
  4. How does the Intermediate Value Theorem rely on continuity?
  5. 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)