Oscillation for almost continuity (Q2460659)

Let \((X,\tau)\) be a topological space. A subset \(A\) of \(X\) is said to be preopen if \(A\subset \text{Int}(Cl(A))\). A function \(f:(X,\tau)\to (Y,\sigma)\) is almost continuous or precontinuous if \(x\in \text{Int}(Cl(f^{-1}(V)))\) for each open set \(V\) containing \(f(x)\). Let \((Y,d)\) be a metric space and \(f:(X,\tau)\to (Y,\tau)\). A function \(a_f(x):X\to\mathbb R\cup\{\infty\}\) is defined as \(a_f(x)=\inf\{\text{diam}\, f(A):A\;\text{is preopen containing}\;x\}\), where \(\text{diam}(M)=\sup\{d(u,v):u,v\in M\}\) is the diameter of a set \(M\). It is proved that \(f\) is almost continuous if and only if \(a_f(x)=0\). Some properties of this function are investigated.