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In Mathematical analysis, the intermediate value theorem is either of two theorems of which an account is given below. more...
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Intermediate value theorem
The intermediate value theorem states the following: If y=f(x) is continuous on , and N is a number between f(a) and f(b), then there is at least one c ∈ such that f(c) = N.
Suppose that I is an interval in the real numbers R and that 
 or
 It is frequently stated in the following equivalent form: Suppose that  < <i>u</i> < <i>f</i>(<i>b</i>) or <i>f</i>(<i>a</i>) > <i>u</i> > <i>f</i>(<i>b</i>). Then for some <i>c</i> ∈ , <i>f</i>(<i>c</i>) = <i>u</i>.</span></p>
<p><span class=) This captures an intuitive property of continuous functions: given f continuous on , if f(1) = 3 and f(2) = 5 then f must be equal to 4 somewhere between 1 and 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting your pencil from the paper.
The theorem depends on the completeness of the real numbers. It is false for the rational numbers Q. For example, the function f(x) = x^2 - 2, x ∈ Q satisfies 
Proof
We shall prove the first case  Let  Read more at Wikipedia.org
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