# Intermediate Value Theorem.

Theorem 5.6.1 (first version; Figure VI1 )   Let be a function, defined on the interval . If is continuous on , then achieves any value between and .

Example 5.6.2   Take , for (see Figure 8).

• The function is the sum of three functions, all of them continuous on the interval ; thus is continuous on , by Theorem cont sum;
• ;
• .
As , by Theorem 6.1, the equation has at least one solution in the interval .

Theorem 5.6.3 (second version; Figure VI2 )   Let be a function, defined on the interval . If is continuous on and if , then vanishes at least once on (i.e. there exists at least one real number such that .

Example 5.6.4   Let .
• The function is continuous on , by Theorem cont sum;.
• .
• .
By Theorem 6.3, there exists at least one number between and such that

Theorem 5.6.5 (generalization; Figure fig VI3)   Let be a function, defined on the interval for which the following conditions hold
(i)
the function is continuous on ;
(ii)
the function has a limit on the right at (eventually infinite);
(iii)
the function has a limit on the left at (eventually infinite);
(iv)
these limts have different signs.
The vanishes at least once on (i.e. there exists at least one real number such that .

Of course this theorem is still valid when the interval of deinition is the whole of .

Corollary 5.6.6   Every polynomial of odd degree has at least one real root.

Proof. Let be a polynomial of odd degree. The function is continuous on and by Proposition prop poly limit, it has infinite limits at and , with opposite signs. The corollary is a direct consequence of Theorem 6.5.

The graph of is diplayed in Figure 10.

Noah Dana-Picard 2007-12-28