Let $x\in X$ which is compact space (for example $S^1$). Suppose $\{f_n(x)\}_{n\geq1}$ is upper semi-continuous, decreasing and $\{g_n(x)\}_{n\geq}$ is lower semi-continuous, increasing, and $f_n(x)\geq g_n(x)$, is it possible to construct a continuous function $h(x)$ in between, that is $f_n(x) \geq h(x)\geq g_n(x)$?

Thank you!