Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 1 / b / ii / Solution

For fixed $h\in G$, the conjugation map is the compositefollowed by multiplication. Inversion, constant maps, diagonal maps, and multiplication are continuous in a topological group, so $c_h(g)=gh{g}^{-1}$ is continuous.