Proof
(related to Theorem: Intermediate Value Theorem)
 Let \([a,b]\) be a closed real interval.
 Let \(f:[a,b]\to\mathbb R\) be a continuous real function.
 We want to shot that \(f\) takes any value between \(f(a)\) and \(f(b)\), i.e. for each \(u\in [f(a),f(b)]\) there is at least one \(c\in[a,b]\) with \(f( c)=u\).
 This is trivial for $u=f(a)$ or $u=f(b)$. Therefor let $f(a) < u < f(b).$
 Set the function $g:[a,b]\to\mathbb R$ with $g(x):=f(x)u$.
 Since $f$ is continuous by hypothesis and the identity function is continuous, $g$ is also continuous, because the arithmetic operation "$$" preserves the continuity.
 Thus we all prerequisites for the intermediate root value theorem are fulfilled and there is a $c\in[a,b]$ with $g(c )=0.$
 It follows that $f(c )=u.$
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983