Prove or give a counterexample: if \(f:\mathbb R\to\mathbb R\) has both a left limit and a right limit at every point, then \(f\) has at most countably many discontinuities.
Solution
For \(x\in\RR\), write \[ f(x-) \coloneqq \lim_{t\to x^-}f(t), \qquad f(x+) \coloneqq \lim_{t\to x^+}f(t), \] and define \[ \Delta(x) \coloneqq \max\left\{ \abs{f(x)-f(x-)}, \abs{f(x)-f(x+)}, \abs{f(x-)-f(x+)} \right\}. \]
1. The function \(f\) is discontinuous at \(x\) if and only if\[ \Delta(x)>0. \]
If \(f\) is continuous at \(x\), then both one-sided limits equal \(f(x)\), so \(\Delta(x)=0\). Conversely, if \(\Delta(x)=0\), then \[ f(x-)=f(x)=f(x+), \] so the two-sided limit exists and equals \(f(x)\). Hence \(f\) is continuous at \(x\).
2. For positive integers \(m,n\), set\[ E_{m,n} \coloneqq \left\{ x\in[-m,m]: \Delta(x)\geq\frac1n \right\}. \]Then \(E_{m,n}\) is finite.
Suppose instead that \(E_{m,n}\) is infinite. By compactness of \([-m,m]\), it has an accumulation point \(c\in[-m,m]\). Choose distinct points \(x_j\in E_{m,n}\) with \(x_j\to c\). After passing to a subsequence, either all \(x_j>c\) or all \(x_j<c\). The two cases are symmetric, so assume \(x_j>c\).
Let \[ L=f(c+). \] By the definition of the right-hand limit, with \[ \eta\coloneqq\frac1{4n}, \] there is \(\delta>0\) such that \[ \abs{f(t)-L}<\eta \] whenever \[ c<t<c+\delta. \] For all sufficiently large \(j\), \[ c<x_j<c+\frac\delta2. \] Thus \[ \abs{f(x_j)-L}<\eta. \] Moreover, all points sufficiently close to \(x_j\) from either side still lie in \((c,c+\delta)\). Passing to the one-sided limits at \(x_j\) gives \[ \abs{f(x_j-)-L}\leq\eta, \qquad \abs{f(x_j+)-L}\leq\eta. \] Therefore every pair among \[ f(x_j), \qquad f(x_j-), \qquad f(x_j+) \] differs by at most \[ 2\eta=\frac1{2n}<\frac1n. \] Hence \[ \Delta(x_j)<\frac1n, \] contradicting \(x_j\in E_{m,n}\).
3. The set \(D\) of discontinuities of \(f\) satisfies\[ D = \bigcup_{m=1}^{\infty} \bigcup_{n=1}^{\infty} E_{m,n}. \]
By step 1, if \(x\in D\), then \(\Delta(x)>0\). Choose \(m\) with \(\abs x\leq m\) and \(n\) with \(1/n\leq\Delta(x)\). Then \(x\in E_{m,n}\). The reverse inclusion follows immediately from step 1.
4. The set of discontinuities of \(f\) is at most countable.
Each \(E_{m,n}\) is finite by step 2, and step 3 expresses \(D\) as a countable union of these finite sets.
5. Thus the proposed statement is\[ \boxed{\text{true}}. \]
Step 4 proves it.
6. Q.E.D.
Step 5 answers the prove-or-counterexample question.