Editing Intermediate value theorem (section)

{{short description|Continuous function on an interval takes on every value between its values at the ends}}
[[File:Illustration for the intermediate value theorem.svg|thumb|Intermediate value theorem: Let <math>f</math> be a continuous function defined on <math>[a,b]</math> and let <math>s</math> be a number with <math>f(a) < s < f(b)</math>. Then there exists some ''<math>x</math>'' between <math>a</math> and <math>b</math> such that <math>f(x) = s</math>.]]
In [[mathematical analysis]], the '''intermediate value theorem''' states that if <math>f</math> is a [[continuous function|continuous]] [[Function (mathematics)|function]] whose [[domain of a function|domain]] contains the [[interval (mathematics)|interval]] {{closed-closed|''a'', ''b''}}, then it takes on any given value between <math>f(a)</math> and <math>f(b)</math> at some point within the interval.

This has two important [[corollary|corollaries]]:

# If a continuous function has values of opposite sign inside an interval, then it has a [[Zero of a function|root]] in that interval ('''Bolzano's theorem''').<ref>{{MathWorld |title=Bolzano's Theorem |urlname=BolzanosTheorem}}</ref><ref>{{cite book |doi=10.1007/978-3-030-11036-9|title=Cauchy's Calcul Infinitésimal |year=2019 |last1=Cates |first1=Dennis M. |isbn=978-3-030-11035-2 |s2cid=132587955|page=249 }}</ref>
# The [[image (mathematics)|image]] of a continuous function over an interval is itself an interval.