In this paper, we introduce a special kind of graph homomorphisms namely semi-locally-surjective graph homomorphisms. We show some relations between semi-locally-surjective graph homomorphisms and colorful colorings of graphs. Then, we prove that for each natural number \(k\), the Kneser graph KG\((2k + 1, k)\) is \(b\)-continuous. Finally, we present some special conditions for graphs to be \(b\)continuous.