Let D D be a bounded pseudoconvex domain with C ∞ {C^\infty } boundary in C n , A ∞ ( D ) {{\mathbf {C}}^n},{A^\infty }(D) the algebra of functions holomorphic in D D and C ∞ {C^\infty } up to the boundary, and M M a compact real-analytic manifold in the boundary which is integral for the complex structure of the boundary and which has no complex tangent vectors. A necessary and sufficient condition that each element of A ∞ ( D ) {A^\infty }(D) be real-analytic on M M is that the germ of the complexification of M M be in the boundary. Examples indicate that the quasi-analyticity of A ∞ ( D ) {A^\infty }(D) along M M is possible even in the absence of complex manifolds in the boundary.