I am learning about representation theory and have some questions to ask. Let $G$ be a compact group. The book "Analysis on Lie groups" by Faraut defines a representation of $G$ on a finite dimensional vector space $V$ to be a continuous map $\pi:G\to GL(V)$ such that $\pi(g_1g_2)=\pi(g_1)\pi(g_2)$ and $\pi(e)=I$.
This continuity assumption is not made in the definition of representation on Wikipedia. Wikipedia only mentions the requirement that $\pi$ is a group homomorphism. Why is this the case?
Second question: There seems to be a relation between group actions and representations according to Wikipedia. However, this again goes by the definition of representation without continuity. I'll explain more below, focussing on one direction.
Suppose that I have a group action $\rho:G\times V\to V$. If $\rho$ is linear, then certainly $\pi(g)(v):=\rho(g,v)$ is linear, and so $\pi:G\to GL(V)$ is well-defined. It is also easy to see that this defines a homomorphism. Hence it is a representation according to Wikipedias definition.
However, what about continuity? Recall the definition in Faraut requires continuity of $\pi$. Do we need an added assumption on the group action (eg continuity)?
Sorry if this is really trivial!
You've neglected the fact that it is to be a representation of a Lie group [i.e a topological group]. The 'homomorphism' part handles the 'representation' aspect, but the 'continuous' part is needed to preserve the 'topological' aspect. If it weren't a topological group, the definition omitting continuity would suffice, as you suggest.