Hahn-Dieudonne-Tong insertion theorem

Statement

Suppose $X$ is a normal space, $f:X\to [0,1]$ is an upper semicontinuous function and $g:X\to [0,1]$ is a lower semicontinuous function. Suppose further than $f\leq g$ pointwise. Then, there exists a continuous function $h:X\to [0,1]$ such that $f\leq h\leq g$.

Conversely, if $X$ is a topological space satisfying the above condition, then $X$ is normal.