Here is the link:
The set of points of continuity of a real-valued function on a metric space is a $G_\delta$ set
But I am not understanding why in the last comment in the last line "$ y,z \in B(\omega, \delta_{\omega})\subseteq B(x, \delta_{x}$)". Could anyone explain this for me please?
It follows by triangle inequality: if $d(y,w) <\delta_w$ then $d(y,x) \leq d(y,w)+d(w,x) <\delta_w+d(w,x)=\delta_x$. Similarly for $z$.
We have taken $x \in U_n$ and we have to show that $x$ is an interior point. It is enough to show that $B(x,\delta_x) \subset U_n$. For this take any point $w \in B(x,\delta_x)$. it is enough to show that $w \in U_n$. By definition of $U_n$ this means there exists $\delta >0$ such that $d(f(y),f(z))<\frac 1 n$ whenever $y, z \in B(w,\delta)$. We take $\delta =\delta_w$ and apply above inequality for arbitrary points $y$ and $z$ in $B(w,\delta)$.