Functions with a finite range are bijective? Suppose we have a function $f : A \subseteq \mathbb{R} \to \{ a_1 , \dots , a_n \} $.
So $\operatorname{range} f $ is finite. Can we conclude that $f$ is bijective? Or only can it be said that it is injective?

Not necessarily. The only way for $f$ to be bijective is that  $A$ was a finite set with $n$ elements.
But $f$ is not bijective in general. For example, the graph of a function with a finite range could be something like this:

and if you use the horizontal line test (red lines in the picture) you will see that the function of this example is not bijective, even if it has a finite range.
