## Quick Reference

The following theorem stating an important property of continuous functions:

Theorem

If the real function *f* is continuous on the closed interval [*a*, *b*] and *η* is a real number between *f*(*a*) and *f*(*b*), then, for some *c* in (*a*, *b*), *f*(*c*)=η.

The theorem is useful for locating roots of equations. For example, suppose that *f*(*x*)=*x*−cos *x*. Then *f* is continuous on [0, 1], and *f*(0)<0 and *f*(1)>0, so it follows from the Intermediate Value Theorem that the equation *f*(*x*)=0 has a root in the interval (0, 1).

*Subjects:*
Mathematics.

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