I was studying continuous functions in metric spaces and I have these doubts

For example

$(x,\tau)$ ; $\tau = \{x,\varnothing \} $

$(y, \mathcal{P}(y))$

For the funtions

$f: x\rightarrow{y}$

$f: y\rightarrow{x}$

$f:x\rightarrow{x}$

what are the conditions for the functions to be continuous?