Originally Posted by

**oblixps** thanks for the proof!

now to prove this i need to show that any preimage of a closed set V in R is closed as well. the closed sets in this case are finite sets and since f is injective, the preimage of f must contain the same number of elements or less than the closed sets in the codomain. so the preimages are finite as well and since a ball around any point in a finite set in R necessarily is not contained in the finite set, the preimages must be closed sets. therefore f is continuous.

would this be the correct reasoning?