Let X ⊂ R^n be a closed semi-algebraic set, F : R^n → R be a C^2 semi-algebraic function and f = F |X : X → R n be the restriction of F to X. We define the global index of a critical value c_i of f and prove an index formula for χ(X) that generalizes a result previously proved by the authors for the case of isolated critical points. We define also new indices at infinity and prove an alternative index formula for χ(X).