Produce a function f(x) that satisfies the following conditions:
I. Its domain is all real numbers.

II. It has no maximum and no minimum on the interval [1, 3].

III. It satisfies f(1) = 1 and f(3) = –1, but there does not exist a c between 1 and 3 such that f(x) = 0.

Question

Produce a function f(x) that satisfies the following conditions:
I.      Its domain is all real numbers.

II.     It has no maximum and no minimum on the interval [1, 3].

III.    It satisfies f(1) = 1 and f(3) = –1, but there does not exist a c between 1 and 3 such that f(x) = 0.

anonymous · Answer

I'm sorry - I forgot to add that you have to use the Intermediate Value Theorem and Extreme Value Theorem.

john_es · Answer

The answer, I think, it is base in the fact that both theorems need the continuity of f(x), so if we build a function without continuity and with those requisites, we would have the solution. An example,
$$f(x)=\begin{cases}2x-1\ \ \ \ \text{if}\ \ \ \  x<2\
2x-7\ \ \ \ \text{if}\ \ \ \  x>2
\end{cases}$$
This function satisfies the conditions imposed by the problem.