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

Analyzing Each Option:

(A) $f$ exists on $(a, b)$:

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

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

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

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

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

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

This is not necessarily true. For example, the function $f(x) = x^2$ is continuous on $[a, b]$, but $f'(x)$ may not be $0$ in the interval unless the specific shape of $f(x)$ demands it.

(E) The graph of $f$ 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 $x_0$?
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.