Which of the following statements are true?
A. If $\lim f ( x )$ exists, then $f$ is continuous at $x = c$.
B. If $f ( 2 ) = - 3$, and $f ( 6 ) = - 8$, then there must be a value $c$ at which $f ( c ) = - 5$.
C. If $f$ is continuous on the closed interval [1, 4], then $f$ is continuous at every point in [1,4].
D. If $f$ is continuous on the interval $( 1,4 )$, then $f$ must have a maximum value and a minimum value on $( 1,4 )$.
A) A and C are true.
B) A, C, and D are true.
C) C and D are true.
D) Only C is true.

Question

Quizplus · Accepted Answer

- A is false because the existence of a limit at $x = c$ does not guarantee continuity at $x = c$; $f$ must also be defined at $c$ and $f(c)$ must equal the limit.- B is false because it incorrectly applies the Intermediate Value Theorem without stating that $f$ is continuous on the interval [2, 6].- C is true because continuity on a closed interval implies continuity at every point within that interval.- D is false because a function can be continuous on an open interval without necessarily having a maximum or minimum value on that interval; these extrema are guaranteed only on closed intervals under certain conditions.

Which of the Following Statements Are True? A lim⁡f(x)\lim f ( x )limf(x)

Which of the Following Statements Are True?
A $\lim f ( x )$