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 $\lambda \in G$ be arbitrary element in $G$.

Let $f,g : \struct{S, \tau_S} \to \struct{G, \tau_G}$ be continuous mappings.

Then the following results hold: