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In this note, we get the conditions such that strong continuity ${\Rightarrow}$ weak continuity plus interiority condition( wc+ic), and continuity ${\Rightarrow}$ wc+ic are true. And we investigate some equivalent conditions with weak continuity, some properties of weak continuity. And we show that almost compactness is preserved by weakly continuous function, and we improve some known results with respect to strong continuity.
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