Given a function $f : A \to B$ and $C \subseteq A$, when $f|_C$ is bijective, then is $f$ bijective?
My initial assumption would be yes, since the domain of the restriction is a subset of the original one, meaning it should be a portion of the original's domain, and if the restriction of the function is bijective, then surely the original one is bijective. However, I am not certain about this, and I would appreciate if somebody helps in clearing this up. Our professor never discussed this so I am not sure if I missed some points or not.
Best Answer
Not necessarily, you can lose injectivity.
For example, $x^2$ is bijective on $[0,1]$ but not on $[-1,1].$