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In this paper, we will characterize the Baire-1 functions f : X -> Y by their graph: namely, we will show that f is a Baire-1 function if and only if its graph gr(f) is the intersection of a sequence (G(n))(n=1)(infinity) of open sets in X x Y such that for all x is an element of X and n is an element of N the vertical section of G(n) is a convex set, whose diameter tends to 0 as n -> infinity. Afterwards, we will discuss a similar question concerning functions of higher Baire classes and formulate some generalized results in slightly different settings: for example we require the domain to be a metrized Suslin space, while the codomain is a separable Frechet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces. 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