Hahn-Dieudonne-Tong insertion theorem

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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.