Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

This has two important corollaries: 1) If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). 2) The image of a continuous function over an interval is itself an interval.

This captures an intuitive property of continuous functions: given f continuous on [1, 2] with the known values f(1) = 3 and f(2) = 5. Then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 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 intermediate value theorem states the following.

Consider an interval ${\displaystyle I=[a,b]}$ in the real numbers ${\displaystyle \mathbb {R} }$ and a continuous function ${\displaystyle f:I\to \mathbb {R} }$. Then,

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