How to study the effect of eps in sklearn.cluster.DBSCAN?
The meaning of $\epsilon$ is that of the neighbourhood size. The neighbourhood of a point $p$, denoted by $N_{\epsilon}(p)$, is defined as the $N_{\epsilon}(p) = \{q \in D | dist(p,q) \leq \epsilon \}$. Here $D$ is a database of $n$ objects (points) and $q$ a query point.
So what you Professor probably wants you to do is to evaluate goodness of clustering algorithm (DBSCAN) based on its input Parameters Epsilon. This goodness of fit you can measure for example with Silhouette score