Question

Let X and Y be a first countable spaces. Prove that f:X$\rightarrow$Y is continuous if whenever x_{n$\rightarrow$}x in X then f(x_n
)$\rightarrow$f(x) in Y

Answer 1

Suppose that san] ix nequence in x , ana as neco implies flan) f(x). Now we will have to prove that of in continuous at a. since x esa first countable space this implies there existe a local base {vnine at x such that unta cun for all nenowe will we method of proof of contradiction of f is not. continuous at x then there should exist an open set w. containing f(x) such that fluje w for any open set u containing x. In particular for such an open set we flun) e w for all n = 1, 2, 3 ... Hence there exista ant Vn such that flan) &w claim! anon as nos . So start with an open set u enx containing x Now ſun is a local base at a implies there exinto no En such that no cv hence ane une nocv for all nano. That is for each open set u containing a there exists no CN such that aut v for all nano. Hence xnx as nos . But this sequence san} en y es auch that flan) & w , where w is an open xet containing f(x). so we have arrived at a contradiction to our cessumption namely anex, xnxex implies flan) f(x) We arrived at this contradiction by assuming f is not continuous at x. Therefore our assumption is wrong and hence of is continuous at x.