In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and there are at most countably many of them.
It is called Froda's theorem in some recent works; in his 1929 dissertation, Alexandru Froda stated that the result was previously well-known and had provided his own elementary proof for the sake of convenience.
[1] Prior work on discontinuities had already been discussed in the 1875 memoir of the French mathematician Jean Gaston Darboux.
be a real-valued monotone function defined on an interval
With this remark the theorem takes the stronger form: Let
This proof starts by proving the special case where the function's domain is a closed and bounded interval
This completes the proof of the special case where the function's domain is a closed and bounded interval.
is associated with a unique rational number (said differently, the map
(a monotone real-valued function) is equal to a union of countably many closed and bounded intervals; say its domain is
To make this argument more concrete, suppose that the domain of
with the property that any two consecutive intervals have an endpoint in common:
Let x1 < x2 < x3 < ⋅⋅⋅ be a countable subset of the compact interval [a,b] and let μ1, μ2, μ3, ... be a positive sequence with finite sum.
Set where χA denotes the characteristic function of a compact interval A.
Then f is a non-decreasing function on [a,b], which is continuous except for jump discontinuities at xn for n ≥ 1.
In the case of finitely many jump discontinuities, f is a step function.
[8][9] More generally, the analysis of monotone functions has been studied by many mathematicians, starting from Abel, Jordan and Darboux.
The main task is to construct monotone functions — generalising step functions — with discontinuities at a given denumerable set of points and with prescribed left and right discontinuities at each of these points.
Let xn (n ≥ 1) lie in (a, b) and take λ1, λ2, λ3, ... and μ1, μ2, μ3, ... non-negative with finite sum and with λn + μn > 0 for each n. Define Then the jump function, or saltus-function, defined by is non-decreasing on [a, b] and is continuous except for jump discontinuities at xn for n ≥ 1.
[10][11][12][13] To prove this, note that sup |fn| = λn + μn, so that Σ fn converges uniformly to f. Passing to the limit, it follows that if x is not one of the xn's.
[10] Conversely, by a differentiation theorem of Lebesgue, the jump function f is uniquely determined by the properties:[14] (1) being non-decreasing and non-positive; (2) having given jump data at its points of discontinuity xn; (3) satisfying the boundary condition f(a) = 0; and (4) having zero derivative almost everywhere.
Property (4) can be checked following Riesz & Sz.-Nagy (1990), Rubel (1963) and Komornik (2016).
Without loss of generality, it can be assumed that f is a non-negative jump function defined on the compact [a,b], with discontinuities only in (a,b).
Note that an open set U of (a,b) is canonically the disjoint union of at most countably many open intervals Im; that allows the total length to be computed ℓ(U)= Σ ℓ(Im).
Note that Uc(f) consists the points x where the slope of h is greater that c near x.
By definition Uc(f) is an open subset of (a, b), so can be written as a disjoint union of at most countably many open intervals Ik = (ak, bk).
By compactness, there are finitely many open intervals of the form (s,t) covering the closure of Jk.
On the other hand, it is elementary that, if three fixed bounded open intervals have a common point of intersection, then their union contains one of the three intervals: indeed just take the supremum and infimum points to identify the endpoints.
To prove this, define a variant of the Dini derivative of f. It will suffice to prove that for any fixed c > 0, the Dini derivative satisfies Df(x) ≤ c almost everywhere, i.e. on a null set.
It follows that Df = g' +Dh = Dh except at the N points of discontinuity of g. Choosing N sufficiently large so that Σn>N λn + μn < ε, it follows that h is a jump function such that h(b) − h(a) < ε and Dh ≤ c off an open set with length less than 4ε/c.
By construction Df ≤ c off an open set with length less than 4ε/c.