Let \((X, d)\) be a metric space, let \(f\colon X\to\RR\), and let \(D_f\subseteq X\) be the set of points at which \(f\) is not continuous. For \(x\in X\), the oscillation of \(f\) at \(x\) is \[ \omega_f(x)\coloneqq\inf_{\delta>0}\,\sup\theset{\abs{f(y) - f(z)} \suchthat y, z\in B_\delta(x)}\in[0,\infty]. \] Then \(f\) is continuous at \(x\) if and only if \(\omega_f(x) = 0\), each set \(\theset{x\in X\suchthat \omega_f(x)\geq 1/k}\) is closed, and \[ D_f = \bigcup_{k\geq 1}\theset{x\in X\suchthat \omega_f(x)\geq 1/k}. \] In particular, \(D_f\) is an $F_\sigma$ set.
If \(f\) is continuous at \(x\) and \(\varepsilon>0\), choose \(\delta>0\) with \(\abs{f(y)-f(x)}<\varepsilon\) for \(y\in B_\delta(x)\); then \(\abs{f(y)-f(z)}<2\varepsilon\) for \(y,z\in B_\delta(x)\), so \(\omega_f(x)\leq 2\varepsilon\). Conversely, if \(\omega_f(x) = 0\) and \(\varepsilon>0\), some \(\delta>0\) has \(\abs{f(y)-f(x)}<\varepsilon\) for all \(y\in B_\delta(x)\). This proves the characterization of continuity, and hence the formula for \(D_f\).
For closedness, suppose \(\omega_f(x)<1/k\), and choose \(\delta>0\) with \(s\coloneqq\sup\theset{\abs{f(y)-f(z)}\suchthat y,z\in B_\delta(x)}<1/k\). For \(x'\in B_\delta(x)\), the ball \(B_{\delta - d(x,x')}(x')\) is contained in \(B_\delta(x)\), so \(\omega_f(x')\leq s<1/k\). Hence \(\theset{\omega_f<1/k}\) is open.
Let \(X\) be a metric space, and let \(f_n\colon X\to\RR\) be continuous functions converging pointwise to \(f\colon X\to\RR\). Then \(D_f\) is meager in \(X\).
(Lebesgue criterion)
Let \(a<b\) and let \(f\colon[a,b]\to\RR\) be bounded. Then \(f\) is Riemann integrable on \([a,b]\) if and only if \(D_f\) has Lebesgue measure \(0\).
Let \(I\subseteq\RR\) be an interval and let \(f\colon I\to\RR\) be monotone. Then \(D_f\) is countable, and \(f\) is differentiable at almost every point of \(I\).
We prove countability; differentiability almost everywhere is Lebesgue’s differentiation theorem for monotone functions. Assume \(f\) is nondecreasing. At each interior point \(x\) of \(I\) the one-sided limits \(f(x^-)\leq f(x)\leq f(x^+)\) exist, and \(f\) is discontinuous at \(x\) if and only if \(f(x^-)<f(x^+)\). For interior points \(x<x'\) of \(D_f\), \(f(x^+)\leq f(x'^-)\), so the open intervals \((f(x^-), f(x^+))\) for interior \(x\in D_f\) are pairwise disjoint. Choosing a rational number in each gives an injection of the interior points of \(D_f\) into \(\QQ\), and \(I\) has at most two endpoints.