A compact space X is called pi-monolithic if for any surjective continuous mapping f : X -> K where K is a metrizable compact space there exists a metrizable compact space T subset of X such that f(T) = K. A topological space X is Baire if the intersection of any sequence of open dense subsets of X is dense in X. Let C-p(X, Y) denote the space of all continuous Y-valued functions C(X,Y) on a Tychonoff space X with the topology of pointwise convergence. In this paper we have proved that for a totally disconnected space X the space C-p(X,{0,1}) is Baire if, and only if, C-p(X, K) is Baire for every pi-monolithic compact space K.
For a Tychonoff space X the space Cp(X,R) is Baire if, and only if, C-p(X,L) is Baire for each Frechet space L.
We construct a totally disconnected Tychonoff space T such that C-p(T, M) is Baire for a separable metric space M if, and only if, M is a Peano continuum. Moreover, C-p(T,[0,1]) is Baire but C-p(T,{0,1}) is not.