**Abstract** : 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.