No continuous transformation $f([a,b])= ]a,b[$

Question

$ a,b\in\mathbb R$ with $a<b $.

Now I want to show that there is NO continuous transformation $f: [a,b] \to \mathbb R $ with $f([a,b])= ]a,b[$

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How can I proove that this transformation don't exist?

Answer 1

As suggested in the comments, the continuous image of a compact set is compact, and $]a,b[$ is not compact. For a more elementary proof, supposed such an $f$ existed. Then the function $[a,b]\to \mathbb{R}$, $x\mapsto \frac{1}{f(x)-a}$ is continuous and unbounded, but a function from $[a,b]$ to $\mathbb{R}$ must be bounded.

Answer 2

$f$ is continuous and $[a,b]$ is compact, so that $f([a,b])$ is also compact.

Answer 3

By the extreme value theorem, $f$ must attain a maximum $M$ on $[a,b]$, which by hypothesis is strictly smaller than $b$. Hence the number $\frac{M+b}{2}$ is not in the image of $f$.