zeroth homotopy group, induced map let $f\colon M\rightarrow N$ be a continuous map between topological spaces.
Then there is a induced map $f_*\colon \pi_0(M)\rightarrow \pi_0(N)$.
I view $\pi_0(M)$ as the set of path connected components of $M$.
Is $f_*$ defined like that:
Let $X$ be a path connected component of $M$, then $f_*(X):=f(X)$ ?
It makes sense, since the continuous image of a path connected space is path connected, but I didn't find a reference for that.
 A: You are on the right track, but $f_*(X)$ should rather be the path component containing $f[X]$. Then it's easy to check that this function is well-defined.
In situations like these where we assign to an object of some category (here topological spaces) an object of another category (here sets) and to a map between two objects a map between their images, we typically think about the following:
Consider map $g:N\to L$. We can compose with $f$ and then consider the induced function $(gf)_*:\pi_0(M)\to π_0(L)$. ON the other hand, we can compose the induced functions $g_*\circ f_*:π_0(M)→π_0(N)→π_0(L)$. We see that both functions are the same. Moreover, the identity map on $M$ induces the identity on $π_0(M)$. So the assignment $\pi_0$ respects the composition of maps. When this holds, we say that $\pi_0$ is a functor, in this case from $\mathbf{Top}$ to $\mathbf{Set}$.
In homotopy theory we consider pointed spaces $(X,x_0)$ and maps preserving the base point $x_0$. These form a category $\mathbf{Top_*}$, and if we take the path component containing $x_0$ as the base point for $\pi_0(X,x_0)$, we obtain a functor $\mathbf{Top_*}→\mathbf{Set_*}$. This subtlety adds some extra flavour to the long exact sequence of relative homotopy groups, but that's a different story.
