Editing Intermediate value theorem (section)

==Theorem==
The intermediate value theorem states the following:

Consider the closed interval <math>I = [a,b]</math> of real numbers <math>\R</math> and a continuous function <math>f \colon I \to \R</math>. Then

*''Version I.'' if <math>u</math> is a number between <math>f(a)</math> and <math>f(b)</math>, that is, <math display="block">\min(f(a),f(b))<u<\max(f(a),f(b)),</math> then there is a <math>c\in (a,b)</math> such that <math>f(c)=u</math>.
*''Version II.'' the [[Image of a function|image set]] <math>f(I)</math> is also a closed interval, and it contains <math>\bigl[\min(f(a), f(b)),\max(f(a), f(b))\bigr]</math>.

'''Remark:''' ''Version II'' states that the [[Set (mathematics)|set]] of function values has no gap. For any two function values <math>c,d \in f(I)</math> with <math>c < d</math> all points in the interval <math>\bigl[c,d\bigr]</math> are also function values, <math display="block">\bigl[c,d\bigr]\subseteq f(I).</math>
A subset of the real numbers with no internal gap is an interval. ''Version I'' is naturally contained in ''Version II''.