Let a function be defined as $f:(\Omega_1,\mathcal{F}_1)\rightarrow (\Omega_2,\mathcal{F}_2)$, where $\mathcal{F}_1$ and $\mathcal{F}_2$ are $\sigma$-fields in $\Omega_1$ and $\Omega_2$ respectively.
If $f$ is continuous then for all $F^1_n \uparrow F^1$, $F^1_n \;\in \mathcal{F}_1$, $$f(F^1_n) \rightarrow f(F^1)$$
i.e. $$\lim_n \sup f(F^1_n) = \lim_n \inf f(F^1_n) = \lim_n f(F^1_n).$$
What I want to know is, whether there is any other standard definition of continuity for these functions i.e. the other definition holds iff this definition holds.
For e.g. in metric spaces the following definitions are equivalent,
- $x_n \rightarrow x$ $\Leftrightarrow f(x_n) \rightarrow f(x)$
- $f^{-1}(A)$ for every open set $A$ is open
My question is motivated from the definition of continuity in section 1.5.1 in Robert Ash.
