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 $$f \le g$$ pointwise. Then, there exists a continuous function $$h:X \to [0,1]$$ such that $$f \le h \le g$$.

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