Continuity of topological spaces

Question

So in this exact problem $f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$. My question is, what if the space given does not mention anything like say space $Y$ is not given nor given as being Hausdorff, will the statement still hold?

Answer 1

Some condition is needed. If $Y$ is an indiscrete space, then every function $f:X\to Y$ is continuous, so one cannot hope to define such a function by continuity if one only knows its values on a proper subset.