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What is the difference between an injective function and a monotonic function? An injection is a function where its values only can be occurred once ($f(a)=f(b) \Rightarrow a=b$). This means that a function is either decreasing or increasing. Isn't this the same for a monotonic function?

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    $\begingroup$ Think about discontinuous functions. $\endgroup$ Jan 15 '13 at 23:42
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    $\begingroup$ There is a theorem which states that, for continuous real functions of a real variable, injectivity and monotonicity are the same. This is what is confusing you. $\endgroup$ Jan 15 '13 at 23:51
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No, these aren't the same. One thing is that injectivity makes sense for functions between arbitrary sets, but for monotonicity you need an order on the domain and codomain.

On the other hand, these concepts are different even for functions $\mathbb{R}\to\mathbb{R}$. For example, $f:\mathbb{R}\to \mathbb{R}$, $f(x)=\begin{cases}x^{-1}&\mbox{if $x\ne0$}\\0&\mbox{if $x=0$}\end{cases}$ is injective but not monotonic.

I guess you have to look again at the definition of monotonic. There are two cases: monotonically increasing, or monotonically decreasing. A monotonic function is one or the other, but not a mixture of the two.

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