Why is the set of points where a function f defined on a metric is continuous is the intersection of countable number of open sets? Can anyone give me any idea?
Here is what I have come up with. Let $S(n) = \{ |f(x)-f(y)| < \ 1/n \ and \ \rho(x, y)<\delta_n\}$ If we can prove that S(n) is open. Then $\cap_{n=1}^{\infty}s(n)$ is the set of points where f is continous. But how to prove that S(n) is open?