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?

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 $f^*(T_2)$.
