Intermediate value theorem

The intermediate value theorem says that if a function, ${\displaystyle f}$, is continuous over a closed interval ${\displaystyle [a,b]}$, and is equal to ${\displaystyle f(a)}$ and ${\displaystyle f(b)}$ at either end of the interval, for any number, c, between ${\displaystyle f(a)}$ and ${\displaystyle f(b)}$, we can find an ${\displaystyle x}$ so that ${\displaystyle f(x)=c}$.

This means that if a continuous function's sign changes in an interval, we can find a root of the function in that interval. For example, if ${\displaystyle f(1)=-1}$ and ${\displaystyle f(2)=2}$, we can find an ${\displaystyle x}$ in the interval ${\displaystyle [1,2]}$ that is a root of this function, meaning that for this value of x, ${\displaystyle f(x)=0}$, if ${\displaystyle f}$ is continuous. This corollary is called Bolzano's theorem.^[1]

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