The authors study a class of multi-dimensional backward stochastic differential equations (BSDEs) of the following form: \[ Y_t = g(X)_T+\int_t^T f(r,X,Y_r,Z_r)\,dr- \int_t^T Z_r\,dW_r,\quad t\in[0,T], \tag{1} \] where \(X\) is an \(n\)-dimensional diffusion satisfying the SDE \[ X_t=x+\int_0^t b(r,X_r)\,dr + \int_0^t\sigma(r,X_r)\,dW_r,\quad t\in[0, T], \] in which \(b: [0, T] \times \mathbb R^n\mapsto \mathbb R^n\), \(\sigma: [0, T] \times \mathbb R^n \times\mathbb R^d\mapsto\mathbb R^{n\times d}\) are some measurable functions, \(f:[0,T] \times C([0, T];\mathbb R^n) \times \mathbb R^m \times \mathbb R^{m\times d}\mapsto \mathbb R^m\) is a `non-anticipative functional' with respect to \(X\), and \(g\) is some functional defined on the path space \(C([0,T];\mathbb R^n)\), \(W=\{W_t;t\geqslant 0\}\) is a standard \(d\)-dimensional Brownian motion. An adapted solution to the BSDE (1) is a pair of \(\mathbf F\)-adapted, \(\mathbb R^m \times\mathbb R^{m\times d}\)-valued processes \((Y,Z)\) that satisfies (1) almost surely. The authors are interested in the following two long-standing problems in the theory of BSDEs: (i) Suppose \(m > 1\), and that the generator \(f\) is only bounded and continuous (in all variables). Do we still have the existence of the (strong) adapted solution to the BSDE (1)? (ii) To what extent we can still have the `nonlinear Feynman-Kac' formula? That is, we can represent an adapted solution of BSDE, whenever it exists, as some function or functional of the forward diffusion via a solution of a system of partial differential equations (PDEs)? This paper is a first attempt to answer these two questions. The authors consider the case where the functionals \(g\) and \(f\) are of the following `discrete-functional type': \[ g(X) = g(X_{t_1},\dots,X_{t_N}),\quad f(t,X, Y_t,Z_t) = f(t,X_{t_1\wedge t},\dots,X_{t_N\wedge t}, Y_t,Z_t), \] where \(0 = t_0 < t_1 <\dots < t_N= T\) is a given partition of \([0, T]\). The authors first prove that the Feynman-Kac formula still holds in this case and derive the corresponding PDEs, in both classical sense and viscosity sense. It is worth noting that in this `piecewise Markovian' case, the authors show that the following representation holds: \[ Y_t = u(t,X_{t_1\wedge t},\dots,X_{t_N\wedge t});\quad Z_t = v(t,X_{t_1\wedge t},\dots,X_{t_N\wedge t})\,\sigma(t,X_t),\quad t\in[0,T], \] where \(u\) and \(v\) are a solution (in a certain sense) of a system of semilinear PDEs. The authors then prove the existence of the adapted solution to BSDE (1) with continuous coefficients in this piecewise Markovian case.