Explain what is wrong with this reasoning:

If f(x)=1/x then

f(1)=1<0 and f(1)=1>0

so f must have a root between x=1 and x=

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1 Answer

The error in the reasoning is the assumption of continuity. f(x) must be continuous in the interval f(a) to f(b) in order to apply the Intermediate Value Theorem (IVT). The definition of continuity is that, if you were to graph f(x), you must be able to do so without having to lift the pencil from the paper between the limits. f(x)=1/x is discontinuous at x=0 and the limit as x➝0 cannot be defined.

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