Cellular homotopy theorem

Statement
Suppose $$(X,A)$$ and $$(Y,B)$$ are relative CW-complexes (i.e. CW-pairs). Let $$f:(X,A) \to (Y,B)$$ be a continuous map. Then there exists a cellular map $$g:(X,A) \to (Y,B)$$ such that $$f$$ and $$g$$ are homotopic relative to $$A$$ (that is, the homotopy does not change the restriction of the function to $$A$$ at all).