Question

If g(x) is continuous for all real numbers and g(3) = -1, g(4) = 2, which of the following are necessarily true?
I. g(x) = 1 at least once
II. lim g(x) = g(3.5) as x aproaches 3.5.
III. lim g(x) as x approaches 3 from the left = lim g(x) as x approaches from the right

A. I only
B. II only
C. I and II only
D. I, II, and III
E. None of these.

I want to say that the answer should be D. I, II, and III since the function is continuous for all real numbers. Would this be correct?

Answer 1

Yes that is correct because if it's continuous then it must pass through 1 and the limit must exist at 3.5 and the limit must equal the same from both sides. So the answer is D (I,II,II)