| Intermediate Value Theorem |
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INTERMEDIATE VALUE THEOREM The intermediate value theorem states the following: Suppose that ''I'' is an Function . Then the image set ''f'' ( ''I'' ) is also an interval, and either it contains ''f''(''b'') , or it contains ''f''(''a'') . I.e.
or
It is frequently stated in the following equivalent form: Suppose that ''f'' : ''b'' → R is continuous and that ''u'' is a real number satisfying ''f'' (''a'') < ''u'' < ''f'' (''b'') or ''f'' (''a'') > ''u'' > ''f'' (''b''). Then for some ''c'' in (''a'', ''b''), ''f''(''c'') = ''u''. This captures an intuitive property of continuous functions: given ''f'' continuous on 2 , 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 can be drawn without lifting your pencil from the paper. The theorem depends on the completeness of the Real Number s. It is false for the Rational Number s Q. For example, the function ''f'' (''x'') = ''x''2-2 from Q to Q satisfies ''f'' (0) = -2, ''f'' (2) = 2. However there is no rational number ''x'' such that ''f'' (''x'') = 0. Proof We shall prove the first case ''f'' (''a'') < ''u'' < ''f'' (''b''); the second is similar. Let ''S'' = {''x'' in b : ''f''(''x'') ≤ ''u''}. Then ''S'' is non-empty (as ''a'' is in ''S'') and bounded above by ''b''. Hence by the Completeness property of the real numbers, the Supremum ''c'' = sup ''S'' exists. We claim that ''f'' (''c'') = ''u''. | ||
| Suppose Next That ''f'' (''c'') < ''u'' Again, By Continuity, There Is A δ > 0 Such That ''f'' (''x'') - ''f'' (''c'') < ''u'' - ''f'' (''c'') Whenever ''x'' - ''c'' < δ Then ''f'' (''x'') < ''f'' (''c'') + ( ''u'' - ''f'' (''c'') ) | ''u'' for ''x'' in ( ''c'' - δ, ''c'' + δ) and there are numbers ''x'' greater than ''c'' for which ''f'' (''x'') < ''u'', again a contradiction to the definition of ''c'' |
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