Editing Continuous function (section)

====Intermediate value theorem====
The [[intermediate value theorem]] is an [[existence theorem]], based on the real number property of [[Real number#Completeness|completeness]], and states:

:If the real-valued function ''f'' is continuous on the [[Interval (mathematics)|closed interval]] <math>[a, b],</math> and ''k'' is some number between <math>f(a)</math> and <math>f(b),</math> then there is some number <math>c \in [a, b],</math> such that <math>f(c) = k.</math>

For example, if a child grows from 1&nbsp;m to 1.5&nbsp;m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25&nbsp;m.

As a consequence, if ''f'' is continuous on <math>[a, b]</math> and <math>f(a)</math> and <math>f(b)</math> differ in [[Sign (mathematics)|sign]], then, at some point <math>c \in [a, b],</math> <math>f(c)</math> must equal [[0 (number)|zero]].