Find a function $f$ such that $f$ is discontinuous but $W(f,x)$ is a continuous function

Question

I need some help

Find a function $f$ such that $f$ is discontinuous but $W(f,x)$ is a continuous function (where $W(f,x)$ is the oscillation of f in a point, this means that $W(f,x)=inf[M_\delta(f)-m_\delta(f)$ | $\delta > 0]$

At first, I thought that the Dirichlet function could help since it is not continuous but I want to prove that $W(f,x)=0$ so this means the oscillation is continuous, some advice, please

Answer 1

For the Dirichlet function $f := 1_{\Bbb{Q}}$, $W(f, x) = 1$ for all $x$. Since for this function, $W(f, x)$ is constant and equal to $1$ for all $x$, $W(f, x)$ is continuous for all $x$.