Combination Theorem for Continuous Mappings/Topological Group

Theorem
Let $\struct{S, \tau_{_S}}$ be a topological space.

Let $\struct{G, *, \tau_{_G}}$ be a topological group.

Let $\lambda \in G$.

Let $f,g : \struct{S, \tau_{_S}} \to \struct{G, \tau_{_G}}$ be continuous mappings.

Then the following results hold: