Proof

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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Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983