On extremal values of continuous monotone functions (Q908539)

Let X be a \(T_ 3\)-space without isolated points. Suppose that for each \(x\in X\) there is a base \({\mathcal B}(x)\) of open neighborhoods of x such that for each \(B\in {\mathcal B}(x)\) the sets B,X-B are connected and Fr B is compact (where Fr T\(=ClT-Int T)\). Let f:X\(\to R\) be a monotone function with the Darboux property. Then f has a strict absolute extremum at any point a where f has a strict relative extremum.