Decide if the following claims are true or false providing e

Decide if the following claims are true or false, providing either a short proof or counterexmaple to justify each conclusion. Assume throughout that g is defined and continuous on all of R.

1. if g(x) >= 0 for all x = 0 as well.

2. if g(r) = 0 for all r in Q, then g(x) = 0 for all x in R.

3. if g(x_0) > 0 for a single point x_0 in R, then g(x) is in fact strictly positive for uncountable many points.

Solution

2)

for every irrational x there exists sequence of rational numbers{x_n} converging to . As g(x_n) = 0 for all n. By continuity g(x) = 0 . Hence g(x) = 0 for all irrationals and hence for all x in R

3)

from continuity for every e>0 there exists d such that for all x in (x_0 - d, x_0 + d) , |g(x) - g(x_0)| <e, Choose e < g(x_0) => for all x in (x_0 - d , x_0 +d) g(x) > 0 => g(x) >0 for uncountable many points.

1) its not clear