The role of topology in continuity Suppose we have two sets $M$ and $N$ endowed with topologies $T_1$ and $T_2$ respectively. Consider a (continuous) map $L: M\to N$. Now if it is possible that we define another topology on M in such a way that the same function becomes discontinuous( is it even possible?), what role did the topology play in the continuity? Is continuity an intrinsic part of the underlying set or it depends on the topology that we define on the set?

Continuity depends on the topology. For example, if N has the trivial topology, or M has the discrete topology, then any map from M to N will be continuous. On the other hand, if N has the discrete topology, then the only continuous functions are locally constant functions. And if M has the trivial topology, the only continuous functions are constant functions EDIT: the only continuous functions are those whose image has the trivial subspace topology, including but not limited to, constant functions.
The rough intuition is that the coarser the topology on M or the finer the topology on N, the "harder" it is for a function from M to N to be continuous.
