Topological center of a left or right topological group

Definition for a left-topological group
Suppose $$G$$ is a left-topological group. The topological center of $$G$$ is defined as the subset of $$G$$ comprising those elements $$g \in G$$ such that right multiplication by $$g$$ is a continuous map from $$G$$ to $$G$$.

Definition for a right-topological group
Suppose $$G$$ is a left-topological group. The topological center of $$G$$ is defined as the subset of $$G$$ comprising those elements $$g \in G$$ such that left multiplication by $$g$$ is a continuous map from $$G$$ to $$G$$.