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?
 A: 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.
A: For a moment let's forget about the topology $T_1$ on $M$, and let's just focus on the function $f : M \to N$ and on the topology $T_2$. 
Let me formulate the following collection of subsets of $M$:
$$f^*(T_2) = \{f^{-1}(U) \mid U \in T_2\}
$$
It's not hard to check that $f^*(T_2)$ is a topology on $M$, and it's immediate that $f$ is continuous with respect to the topology $f^*(T_2)$ on $M$ (and the topology $T_2$ on $N$).
Now, let's bring into the picture any topology $T$ on $M$. We can now conclude easily that all of the following properties are equivalent to each other:


*

*$f$ is continuous with respect to the topology $T$ on $M$ (and the topology $T_2$ on $N$).

*$f^*(T_2) \subset T$.

*$T$ is finer than or equal to $f^*(T_2)$.

*$f^*(T_2)$ is coarser than or equal to $T$.


From all of this, we also get the following conclusion

$f^*(T_2)$ is the coarsest topology on $M$ with respect to which $f$ is continuous.

So, to summarize and to answer your question: Yes, if we choose a new topology $T$ on $M$ we can determine whether or not $f$ is still continuous with respect to $T$, by examining the above equivalent conditions. However, the answer doesn't have much to do with the original topology $T_1$ that was given on $M$. Instead the answer is determined by the given topology $T_2$ on $N$, and the given map $f$, and the relationship between $T$ and $f^*(T_2)$.
A: Topologies on a space are related by how many open sets they contain. A topology $\tau_1$ on $M$ can be coarser $\tau_1\subset\tau_2$ or finer $\tau_2\subset\tau_1$ than another topology $\tau_2$ on M.
If we take the coarsest topology on $M$, the discrete topology $\tau_D$ on $M$ every map is continuous. The coarser the topology on M the more maps are continuous.
On the other hand the finer the topology on $N$ the easier it is to find a topology on $M$ that the function is continuous.
The relation to the separation axioms is more complicated. There are some easy results like $\tau_D$ is always T2, but general results strongly depend on the space to get results of the topology on the space.
A: The very definition of continuity says that $f$ is continuous iff, for all open subsets $U\subseteq N$, the preimage $f^{-1}(U)$ is an open subset of $M$.
The very definition of a topology on $N$ (resp. $M$) is that it tells us which subsets of $N$ (resp. $M$) are open.
From here it should be crystal clear that whether $f$ is continuous depends on what topologies we put on $M$ and $N$, i.e. that the same function $f$ can be continuous for one choice of topologies on $M$ and $N$, but discontinuous for a different choice. Examples are in the other answers.
