If the derivative of $f$ is never zero, then $f$ is one-to-one
If the function is differentiable and $f'(x)\neq0$, by the Darboux Theorem it has the intermediate value property.
Suppose by contradiction that there are two real numbers $x_1,x_2 \in A$ such that $f'(x_1)<0$ and $f'(x_2)>0$. This implies the existence of $x_3\in A$ s.t. $f'(x_3)=0$, contradicting the hypothesis $f'(x)\neq 0$.
Because the derivative $f'$ is either strictly positive or strictly negative on $A$, $f$ is monotonic and hence injective.