EDB — 0KG

[0KG]Prerequisites:[0K5]↺↻,[0KC]↺↻.Let $X,Y$ be Hausdorff topological spaces. Let $f:X\to Y$, $x_0\in X$. The following are equivalent.

1. $f$ is continuous at $x_0$;

2. for each net $\phi :J\to X$ such that

   $$\underset{j\in J}{lim}\phi (j)=x_0$$

   we have

   $$\underset{j\in J}{lim}f(\phi (j))=f(x_0).$$

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 $\phi :J\to X$ with the property that $\phi (U)\in U$ for each $U\in J$; note that $\underset{j\in J}{lim}\phi (j)=x_0$.