Continuity in set functions

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,

1. $x_n \rightarrow x$ $\Leftrightarrow f(x_n) \rightarrow f(x)$
2. $f^{-1}(A)$ for every open set $A$ is open

My question is motivated from the definition of continuity in section 1.5.1 in Robert Ash.

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These need not be topologies on $\mathcal{F}_i$, $i=1,2$ even if you defined all convergent sequences. These particularly may not be topologies if $\mathcal{F_i}$ is atomless in the sense of each $F\in \mathcal{F}_i$ contains a measurable non-empty strict-subset. –  Rabee Tourky Jan 21 '13 at 11:58
@RabeeTourky strictly speaking, $f$ is a correspondence. What is the definition for continuity for $f$, and is there any other equivalent definition. It may be completely different than the above 2 given for metric spaces. –  UnadulteratedImagination Jan 21 '13 at 14:18
I think you want $\Rightarrow$, not $\Leftrightarrow$ in definition "1". –  Yoni Rozenshein Mar 23 '13 at 13:36