In this paper, based on the well-known Wu' method, a new algorithm is presented for finding at least one point on each semi-algebraically connected component of a real algebraic set defined by a polynomial system. Let $F$ be a computable ordered field with real closed extension $R$, let $P$ be a finite subset of the ring $F[x_1,...,x_n]$ of polynomials over $F$ in $n$ variables, and $\mbox{Zero}_R(P)$ the set of zeros of $P$ in $R^n$. Using the so-called rational univariate representations in coding the output points, our algorithm can yield a finite subset $\Delta$ of $\mbox{Zero}_R(P)$ such that $\Delta$ meets each semi-algebraically connected component of $\mbox{Zero}_R(P)$. Moreover, the notion of locally critical points is introduced for a semi-algebraic subset. The technique of this paper is to catch the lacally critical points of real algebraic sets by Wu's method.