Compositions of continuous functions and connected functions

Abstract

Suppose f : X → Y f:X \to Y is continuous and onto and g : Y → Z g:Y \to Z is such that g ∘ f : X → Z g \circ f:X \to Z has a property we are interested in. For which properties of functions can we infer that g g has the same property? Properties for which we can infer this include continuity and the Darboux property. Properties for which we cannot include almost continuity.