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In general there is no continuous function between u.s.c. $f$ and l.s.c. $g\leq f$. For example, take $f(x)=1, 0\leq x\leq 1;\; f(x)=0, 1<x\leq 2$, this is u.s.c. Now $g(x)=1, 0\leq x<1;\; g(x)=0, 1\leq x\leq 2$, then $g(x)\leq f(x)$ and evidently there is no continuous function in between.