Should every injective function be non monotonous? Is it necessary, that for a function to be injective (one-one) at all points on its domain, it should strictly not be monotonic at any point on its domain?
I reckon that this should be true, because an injective function should have a unique output associated with every input which isn’t valid for monotonous function. Am I correct? Please correct me, wherever wrong. Thanks for your help!
 A: Assuming that $f$ is continuous (a noncontinuous injective function can jump around all it wants) then an injective function must be either strictly increasing or strictly decreasing (because otherwise it would be "doubling back" on itself--- we can make a calculus proof but ... it is essentially just that intuitive argument; if it's decreasing on some interval but increasing on another it must be going over the same points at least twice.)
However you seem to be confused about what "monotonic" means.  That behavior we just described is exactly what monotonic does mean and an injective continuous function must be monotonic.
(Assuming the function is continuous) then monotonic means the derivative does not change signs.  So the derivative is either always non-negative or alway non-positive.  If the derivative is always $\ge 0$ then the function can never be decreasing so we say it is monotonically non-decreasing.  If the derivative is always $\le 0$ then the function can never be increasing so we say it is monotonically non-increasing.
I think you confusion comes from these function do allow for intervals in the domain where the function is "flat", has derivative of $0$, is constant.  These monotonic functions can't be injective. To be injective the function must be of a stronger type of monotony.
If the function is not allowed to have a $0$ derivative for any measurable integer then it is not merely non-decreasing or non-increasing, it is actively increasing or actively decreasing.
If the function is continuous and the derivative is $\ge 0$ and the derivative is never $0$ an a measurable interval of the domain, we conclude the function is monotonically increasing.  Notice monotonicaly increasing $\implies $ monotonically non-decreasing.  Monotonically non-decreasing is a weaker condition and monotonically increasing is a stronger one.
Same are conditions if the derivative is $\le 0$ but never on an interval:  That is called monotonically decreasing.  And monotonically decreasing $\implies$ monotonically non-increasing.
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[Note: this are equivalent consequences of the function being continuous and not actual definitions. ]
