Let $(X,d)$ be a metric space with at most countably-infite many discontinuities and fix $x \in X$. For every $r \geq 0$, define the sets $A(r)$ by $$ A(r)\triangleq \{y \in X : d(x,y)=r\}, $$ and define the function $$ \begin{aligned} g:[0,\infty)\rightarrow [0,\infty)\\ g(r)\triangleq \begin{cases} 1 : & A(r)\neq \emptyset\\ 0 :&A(r)= \emptyset\\ \end{cases}. \end{aligned} $$
Edit How can I prove that the number of discontinuities of $g$ is bounded above by the number of components of $(X,d)$.
Note: The original question asked for equality, but @AIM_BLB showed that is false.