It makes sense for continuity to be defined on a function mapping a real line to a real line. Or how continuity is defined on a function between two topological spaces (every preimage of an open set is an open set). But what does, continuity on a function between a topological space and real line interval mean? Like in Urysohn's Lemma.
Is it just defined in a particular way or one can derive its meaning by looking at how continuity is defined b/w two topological spaces and two real line intervals?
I'm not very good with words so I hope the question made some sense.
Thanks in advance.