CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-11

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A bounded-variation function has at most countably many discontinuities, all of the first kind

Statement

If $f : [a, b] \to R$ has bounded variation, every well-posed one-sided limit of $f$ exists. Consequently every discontinuity is of the first kind, and the set of discontinuities is at most countable.

Facts & Assumptions

Given: A bounded-variation function $f : [a, b] \to R$ .

[L1]

Jordan decomposition writes $f = f (a) + P_{f} - N_{f}$ with $P_{f}, N_{f}$ nondecreasing (Jordan decomposition for functions of bounded variation).

[L2]

Every well-posed one-sided limit of a monotone function exists, so all its discontinuities are of the first kind (Monotone functions have every relevant finite one-sided limit; interior discontinuities are jumps, with side limits ordered by the direction of monotonicity, The left and right limits of $f$ at $c$ , as limits of the restrictions of $f$ to $A \cap (- \infty, c)$ and $A \cap (c, \infty)$ ).

[L3]

The discontinuity set of a monotone function on an interval is at most countable (Froda's theorem: the set of discontinuities of a monotone function on an interval is at most countable, the injection into $N$ being built from one fixed enumeration of the rationals by least index, so no choice principle is used).

[L4]

Finite sums and differences preserve existing finite function limits (Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero).

Proof

technique · direct

1.1L1L2L4

Apply [L2] to $P_{f}$ and $N_{f}$ . At every endpoint or interior point where a one-sided limit is defined, both component limits exist, and [L4] gives the corresponding one-sided limit of $f = f (a) + P_{f} - N_{f}$ . Thus $f$ has no discontinuity of the second kind.

1.2L1L4

If both $P_{f}$ and $N_{f}$ are continuous at a point, [L4] makes $f$ continuous there. Hence the discontinuity set of $f$ is contained in the union of the two component discontinuity sets.

2.1step 1.1step 1.2L3algebra∎

Each component discontinuity set is at most countable by [L3]. Given injections of them into $N$ , map the first set to the even naturals and the points belonging only to the second to the odd naturals; this injects their union into $N$ . Step 1.2 therefore makes the discontinuity set of $f$ at most countable, and step 1.1 makes every one of its discontinuities first-kind.

A bounded-variation function has at most countably many discontinuities, all of the first kind

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources