Is the given function $f$ continuous?

Question

Problem

Let $\mathbb{R}_l$ denote the reals with lower limit topology, and let $\mathbb{R}_l\times \mathbb{R}_l$ have the product topology. Then the map $f:\mathbb{R}_l\times\mathbb{R}_l\to\mathbb{R}_l$ defined by $f(x,y)=xy$ is continuous. (True or False)

I am trying to use Local formulation of continuity (Munkres) which states that $f:X\to Y$ is continuous if $X$ can be written as the union of open sets $U_\alpha$ such that $f|U_\alpha$ is continuous for each $\alpha$. Can it be solved using this? If not then how to solve?

Thanks in advance.

Answer 1

NO. Let $V=[0,1)$. Let $p=<-1,0>\in R_l^2.$ We have $f(p)=0 . $ Suppose $U$ is a nbhd of $p$ such that $$\forall q\in U\; ( f(q)\in V). $$ We have $U\supset [-1,-1+r)\times [0,s)$ for some $r,s \in R^+ . $ But then $q=<-1,s/2>\in U$ and $f(q)=-s/2\not \in V, $ a contradiction.

Good Q nevertheless.