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Bolzano's theorem states that if

is a continuous function in the closed interval

with

and

of opposite sign, then there is a

in the open interval

such that

Snapshot 1: The function is positive in the interval and therefore

for all

Snapshot 2: The function is negative in the interval so

for all

Snapshot 3: The function is positive for

and negative for

, therefore there is a

such that

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Bolzano's Theorem