Limit vs interior definition of continuity Suppose I have two topological spaces $X$ and $Y$ whose topologies are defined by interior operators $\text{int}_X$ and $\text{int}_Y$ respectively, as well as a function $f$ with domain $I$ (for input) and codomain $O$ (for output).

Here's where I hit my quandary: I see two potential routes for defining continuity. I could say the function $f: I \rightarrow O$ is continuous at the point $\tilde{x}$ iff $\lim_{x \rightarrow \tilde{x}} f(x) = f(\tilde{x})$, then say it's continuous everywhere in $I$ if it's continuous at every point $\tilde{x}$ in $I$.

Or, I could say the function $f: I \rightarrow O$ is continuous iff $f^{1-}\left[\text{int}_X(A)\right]$ is a subset of $\text{int}_Y \left( f^{1-}[A] \right)$ for any $A \subseteq O$. My question is: are these two definitions equivalent? Could someone give me some insight into their equivalence or lack thereof? (I'm assuming the limits are defined with respect to the topologies of $\text{int}_X$ and $\text{int}_Y$.)
Following article is giving the proof you're looking for: Continuous functions and convergent sequences
Following article is giving the proof you're looking for: Continuous functions and convergent sequences