EDB — 0KG

[0KG]Prerequisites:[0K5]↺↻,[0KC]↺↻.Let \(X,Y\) be Hausdorff topological spaces. Let \(f:X→ Y\), \(x_ 0\in X\). The following are equivalent.

1. \(f\) is continuous at \(x_ 0\);

2. for each net \(φ:J→ X\) such that

   \[ \lim _{j∈ J} φ(j) = x_ 0 \]

   we have

   \[ \lim _{j∈ J} f(φ(j)) = f(x_ 0)\quad . \]

Hint, for proving that 2 implies 1. Suppose that \(x_ 0\) is an accumulation point. Consider the filtering set \(J\) given by the neighborhoods of \(x_ 0\); consider nets \(φ:J→ X\) with the property that \(φ(U)∈ U\) for each \(U∈ J\); note that \(\lim _{j∈ J} φ(j) = x_ 0\).