Yes, the definition of an injection is wrong. The definition provided seems to be the traditional definition of a standard singular function.
These generally, functions, or real valued maps, are not many valued (which is generally what is meant by function) always satisfy the requirement that every element in the domain is mapped to one and only one element of the range, but not the conversely. they need to preserve distinct-ness or be injective.every domain element attains a distinct element of the co-domain.,
. In-jectivity relates to the fact, that no two elements of the domain ever-correspond to the same element of the co-domain. It preserves distinct-ness
ie (injective-function) $F(x)=F(y) \leftrightarrow x=y$ and every element of the co-domain is corresponds to at most one element of the domain; distinct elements of the domain have a distinct images.
injective-functions are such that every element of the domain corresponds to one and only one element of the co-domain,(which is a property that function have in general) and in addition, in-jective function have this addition property that no function value can be repeated distinct domain values correspond to distinct function values, the function only takes the same function value if it is the domain point, if $x\neq y\to $F(x)\neq F(y);
The restriction of the injective function F$ from its domain to its image:
$F1:\text{dom}(F)\to \text{Im(F)}$
$IM(F)\subseteq \text{co-dom}(F)$ where $y\in\text{Im(F),are those}\, y\in \text{codom}(F)$ ,for which there $\exists x \in \text{dom}(F): F(x)=y
The inverse of this restricted function is defined and as in-jective uniquely so,
That this the restriction that part of the co-domain which correspond to an element of F's domain, they are defined.
which are is be sur-jective by definition, for a function, and thus F1 will be bi-jective , as its also in-jective
But the original function F$ $F:\text{dom}(F)\to\text{co-dom(F)}$
may not be not be surjective, as it may not have defined inverese for every element of the $\text{codom(F)}. some of those points may not correspond to an element of the domain.
, noR will this restriction F1$ be necessarily an interval; otherwise strictly mon-0tone functions which have bi-jective strictly increasing restrictions, would count as continuous,as image would be interval, which is not always so. Their inverse function may have an ill defined domain (not necessarily a closed interval)
whilst a Function generally only has this horn $ x=y \rightarrow F(x)=F(y)$ (and for every element of the domain it is mapped to at one, and only one element of the co-domain) to