Let $X$ and $Y$ be compact sets (both subsets of the real line). Assume $Y$ has a non-empty interior. Consider the continuous function $f:X \rightarrow Y$. For any given $y$ in the interior of Y, and for $h>0$ small enough so that $y+h \in Y$. I want to know whether it is true that $f^{-1}([y,y+h]) \rightarrow f^{-1}(y)$ when $h \rightarrow 0$
I know that $f^{-1}([y,y+h])$ is a closed set for each $h$. My main problem is that we are talking about convergence between two sets. How do you define convergence between two sets? More informally, my question is does the sequence of set converges to the set composed of the inverse of image of $f(y)$
help?