Given a set S of points in the plane and an angle $0 < \theta \le 2\pi$, the continuous Yao graph $cY (\theta )$ with vertex set S and angle $\theta$ defined as follows. For each $p, q \in S$, we add an edge from p to q in $cY (\theta )$ if there exists a cone with apex p and angular diameter $\theta$ such that q is the closest point to p inside this cone.In this paper, we prove that for $0<\theta <\pi /3$ and $t\ge \frac{1}{1-2\sin (\theta /2)}$, the continuous Yao graph $cY(\theta )$ is a $\mathcal {C}$-fault-tolerant geometric t-spanner where $\mathcal {C}$ is the family of convex regions in the plane. Moreover, we show that for every $\theta \le \pi$ and every half-plane h, $cY(\theta )\ominus h$ is connected, where $cY(\theta )\ominus h$ is the graph after removing all edges and points inside h from the graph $cY(\theta )$. Also, we show that there is a set of n points in the plane and a convex region C such that for every $\theta \ge \frac{\pi }{3}$, $cY(\theta )\ominus C$ is not connected.Given a geometric network G and two vertices x and y of G, we call a path P from x to y a self-approaching path, if for any point q on P, when a point p moves continuously along the path from x to q, it always get closer to q. A geometric graph G is self-approaching, if for every pair of vertices x and y there exists a self-approaching path in G from x to y. In this paper, we show that there is a set P of n points in the plane such that for some angles $\theta$, Yao graph on P with parameter $\theta$ is not a self-approaching graph. Instead, the corresponding continuous Yao graph on P is a self-approaching graph. Furthermore, in general, we show that for every $\theta >0$, $cY(\theta )$ is not necessarily a self-approaching graph.