Can we find the boundary condition of a function of diffusion process?

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given $dx_t=b(t,x_t)dt+\sigma(t,x_t)dW_t$, if it is in one dimensional case, one can use Feller non-explosion test to see if $x_t$ attains a particular boundary. How about $f(x_t)$, $f$ is any arbitrage $C^2$ function? Can we still use Feller non-explosion test?

How about if the Brownian motion is of high dimension? Do we still have similar result?