I have seen multiple proofs online that use continuity to prove that the set is connected. But how will we go about doing this without the use of mapping? Is there any way we can do this by using the definition of connectedness (not a union of non-empty disjoint open sets)?
Maybe you can, but I know of no such proof. You need to show that connectedness is preserved by continuous maps anyway (and that proof is quite trivial), to show it is a topological property, and for many applications (intermediate value theorem among others). There is no reason to avoid using it.