A semi-continuous function with a dense set of points of discontinuity

Let’s come back to Thomae’s function which is defined as:
\mathbb{R} & \longrightarrow & \mathbb{R} \\
x & \longmapsto & 0 \text{ if } x \in \mathbb{R} \setminus \mathbb{Q} \\
\frac{p}{q} & \longmapsto & \frac{1}{q} \text{ if } \frac{p}{q} \text{ in lowest terms and } q > 0

We proved here that \(f\) right-sided and left-sided limits vanish at all points. Therefore \(\limsup\limits_{x \to a} f(x) \le f(a)\) at every point \(a\) which proves that \(f\) is upper semi-continuous on \(\mathbb R\). However \(f\) is continuous at all \(a \in \mathbb R \setminus \mathbb Q\) and discontinuous at all \(a \in \mathbb Q\).

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