Prove that the composition of continuous functions is continuous

Problem: Prove that the composition of two continuous functions is continuous.

Proof. With the topological definition, it's easy. The preimage of an open set under the first function is open. The preimage of that set is open under the second function. Therefore the preimage of the original open set is open under the composition of these two functions. QED

So, algebraically, it all comes from: $$(gf)^{-1}(U)=f^{-1}(g^{-1}(U)).$$

If you are using a more elementary (calculus) definition of continuity, this will take more work.

• If composition is continuous, does it mean that the both functions are continuous? Not even one.