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### How can I prove that a continuous injective function is increasing/decreasing? [duplicate]

If I have a continuous, injective function mapping the real numbers, then it is either increasing or decreasing. This seems intuitively obvious but I can't come up with a neat proof for it.
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### Let $f$ be an injective and continous function. Prove that $f$ is monotone. [duplicate]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an injective and continous function. Prove that $f$ is monotone. My proof-trying Since $f$ is injective for all $x_{1},x_{2}\in\mathbb{R}$ there is a $y$ ...
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### A continuous function $f:\Bbb R\to \Bbb R$ is injective if and only if it is strictly increasing or strictly decreasing [duplicate]

Is the following statement true? A continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ is injective if and only if $f$ is strictly increasing or strictly decreasing. If it's true give a ...
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### Hint for real analysis question [duplicate]

Suppose that $f$ is one-to-one and continuous on [$a,b$]. Prove that $f$ is either strictly increasing or strictly decreasing on [$a,b$].
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I've been trying to add math rigor to a solution of the functional equation in [1], eq. (22). It is: $$f\left(\frac{x}{f(x)}\right) = \frac{1}{f(x)}\,,$$ where you know that $f(0)=1$ and $f(-x) = f(... 2answers 2k views ### When is a continuous function also a bijective function? To begin with, I would like to set forth a property of continuous functions: There doesn't exist a continuous function$f$on$\mathbb{R}$such that$f|_{\mathbb{R}\setminus \mathbb{Q}} : \mathbb{...
Let $f:[a,b]\rightarrow[f(a),f(b)]$ be strictly increasing continuous function (i.e $x>y \implies f(x)>f(y)$). Prove that f is invertible. Proving that the function is one-to-one was simple ...