Decide if the following claims are true or false, providing either a short proof or counterexample to justify each conclusion. Assume throughout that g is defined and continuous on all of R. (a) If g(x)\geq 0 for all x<1, then g(1)\geq 0 as well.
(b) If g(r)=0 for all r \in Q, then g(x)=0 for all x \in R.
(c) If g(x0) > 0 for a single point x_{0} \in R, then g(x) is in fact strictly positive for uncountably many points.