Intermediate value theorem

One consequence of a function, f, being continuous on an interval, [a, b], is that if c is a number between f(a) and f(b), then there exists at least 1 number, x, in the interval, [a, b], such that f(x) = c. This result is called the intermediate value theorem. Given that a function, f, is continuous on the interval [a, b], we can trace the graph of f from a to b without making sudden jumps or lifting the pencil. Thus, the intermediate value theorem makes sense in that if f(a) < c < f(b) and we trace the graph from a to b without lifting the pencil, we have to cross the line y = c at some point x where a < x < b. Refer to the figure below for an illustration of the intermediate value theorem:

Note that the graph could cross y = c more than once, as shown below where the intersection points are x1, x2, x3, and x4.

See also continuous.