Let’s come back to **Thomae’s function** which is defined as:

\[f:

\left|\begin{array}{lrl}

\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

\end{array}\right.\]

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 \(a \in \mathbb Q\).