(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let X be the set of continuous real-valued functions on [0,1]. Prove that S={g in X: g(t) > 0 for all t} is an open subset.

2. Relevant equations

3. The attempt at a solution

I was thinking of taking an arbitrary t[itex]_{0}[/itex] in g and another function f that is also continuous on [0,1], then finding a radius r > g(t[itex]_{0}[/itex])-f(t[itex]_{0}[/itex]) and making the open ball or tube or whatever with that radius about the function g. Somehow I feel that this is incorrect though.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Openness of continous real valued functions

**Physics Forums | Science Articles, Homework Help, Discussion**