Continuity

Apr 2010
17
0
Let f be defined and continuous on a closed set S in R. Let A={x: x\(\displaystyle \in\)S and f(x)=0}.
Prove that A is a closed subset of R .
 

Plato

MHF Helper
Aug 2006
22,507
8,664
Let f be defined and continuous on a closed set S in R. Let A={x: x\(\displaystyle \in\)S and f(x)=0}.
Prove that A is a closed subset of R .
Hint: If \(\displaystyle f\) is continues and \(\displaystyle f(p)\not=0\) then there is an open interval such that \(\displaystyle p\in (s,t)\) and \(\displaystyle f\) is non-zero on \(\displaystyle (s,t)\).
Hence, does this show the complement open?